Machine Learning Group Publications

Publications, Machine Learning Group, Department of Engineering, Cambridge

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Gaussian Processes and Kernel Methods

Gaussian processes are non-parametric distributions useful for doing Bayesian inference and learning on unknown functions. They can be used for non-linear regression, time-series modelling, classification, and many other problems.

David Duvenaud, Oren Rippel, Ryan P. Adams, and Zoubin Ghahramani. Avoiding pathologies in very deep networks. In 17th International Conference on Artificial Intelligence and Statistics, Reykjavik, Iceland, April 2014.

Abstract: Choosing appropriate architectures and regularization strategies for deep networks is crucial to good predictive performance. To shed light on this problem, we analyze the analogous problem of constructing useful priors on compositions of functions. Specifically, we study the deep Gaussian process, a type of infinitely-wide, deep neural network. We show that in standard architectures, the representational capacity of the network tends to capture fewer degrees of freedom as the number of layers increases, retaining only a single degree of freedom in the limit. We propose an alternate network architecture which does not suffer from this pathology. We also examine deep covariance functions, obtained by composing infinitely many feature transforms. Lastly, we characterize the class of models obtained by performing dropout on Gaussian processes.

James Robert Lloyd, David Duvenaud, Roger Grosse, Joshua B. Tenenbaum, and Zoubin Ghahramani. Automatic construction and natural-language description of nonparametric regression models. Technical Report arXiv:1402.4304 [stat.ML], February 2014.

Abstract: This paper presents the beginnings of an automatic statistician, focusing on regression problems. Our system explores an open-ended space of possible statistical models to discover a good explanation of the data, and then produces a detailed report with figures and natural-language text. Our approach treats unknown functions nonparametrically using Gaussian processes, which has two important consequences. First, Gaussian processes model functions in terms of high-level properties (e.g. smoothness, trends, periodicity, changepoints). Taken together with the compositional structure of our language of models, this allows us to automatically describe functions through a decomposition into additive parts. Second, the use of flexible nonparametric models and a rich language for composing them in an open-ended manner also results in state-of-the-art extrapolation performance evaluated over 13 real time series data sets from various domains.

Roger Frigola, Fredrik Lindsten, Thomas B. Schön, and Carl Edward Rasmussen. Bayesian inference and learning in Gaussian process state-space models with particle MCMC. In L. Bottou, C.J.C. Burges, Z. Ghahramani, M. Welling, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems 26. Curran Associates, Inc., 2014.

Abstract: State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning in nonlinear nonparametric state-space models. We place a Gaussian process prior over the transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. However, to enable efficient inference, we marginalize over the dynamics of the model and instead infer directly the joint smoothing distribution through the use of specially tailored Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. We make use of sparse Gaussian process models to greatly reduce the computational complexity of the approach.

José Miguel Hernández-Lobato, James Robert Lloyds, and Daniel Hernández-Lobato. Gaussian process conditional copulas with applications to financial time series. In Advances in Neural Information Processing Systems 27, pages 1-9, Lake Tahoe, California, USA, December 2013.

Abstract: The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be inaccurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables. We evaluate the ability of our method to predict time-varying dependencies on several equities and currencies and observe consistent performance gains compared to static copula models and other time-varying copula methods.

David Lopez-Paz, Philipp Hennig, and Bernhard Scholköpf. The randomized dependence coefficient. In Advances in Neural Information Processing Systems 27, pages 1-9, Lake Tahoe, California, USA, December 2013.

Abstract: We introduce the Randomized Dependence Coefficient (RDC), a measure of non-linear dependence between random variables of arbitrary dimension based on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. RDC is defined in terms of correlation of random non-linear copula projections; it is invariant with respect to marginal distribution transformations, has low computational cost and is easy to implement: just five lines of R code, included at the end of the paper.

Michael A. Osborne, David Duvenaud, Roman Garnett, Carl Edward Rasmussen, Stephen J. Roberts, and Zoubin Ghahramani. Active learning of model evidence using Bayesian quadrature. In Advances in Neural Information Processing Systems 25, pages 46-54, Lake Tahoe, California, USA, December 2013.

Abstract: Numerical integration is a key component of many problems in scientiﬁc computing, statistical modelling, and machine learning. Bayesian Quadrature is a model-based method for numerical integration which, relative to standard Monte Carlo methods, offers increased sample efficiency and a more robust estimate of the uncertainty in the estimated integral. We propose a novel Bayesian Quadrature approach for numerical integration when the integrand is non-negative, such as the case of computing the marginal likelihood, predictive distribution, or normalising constant of a probabilistic model. Our approach approximately marginalises the quadrature model’s hyperparameters in closed form, and introduces an active learning scheme to optimally select function evaluations, as opposed to using Monte Carlo samples. We demonstrate our method on both a number of synthetic benchmarks and a real scientiﬁc problem from astronomy.

Tomoharu Iwata, David Duvenaud, and Zoubin Ghahramani. Warped mixtures for nonparametric cluster shapes. In 29th Conference on Uncertainty in Artificial Intelligence, Bellevue, Washington, July 2013.

Abstract: A mixture of Gaussians fit to a single curved or heavy-tailed cluster will report that the data contains many clusters. To produce more appropriate clusterings, we introduce a model which warps a latent mixture of Gaussians to produce nonparametric cluster shapes. The possibly low-dimensional latent mixture model allows us to summarize the properties of the high-dimensional clusters (or density manifolds) describing the data. The number of manifolds, as well as the shape and dimension of each manifold is automatically inferred. We derive a simple inference scheme for this model which analytically integrates out both the mixture parameters and the warping function. We show that our model is effective for density estimation, performs better than infinite Gaussian mixture models at recovering the true number of clusters, and produces interpretable summaries of high-dimensional datasets.

David Duvenaud, James Robert Lloyd, Roger Grosse, Joshua B. Tenenbaum, and Zoubin Ghahramani. Structure discovery in nonparametric regression through compositional kernel search. In 30th International Conference on Machine Learning, Atlanta, Georgia, USA, June 2013.

Abstract: Despite its importance, choosing the structural form of the kernel in nonparametric regression remains a black art. We define a space of kernel structures which are built compositionally by adding and multiplying a small number of base kernels. We present a method for searching over this space of structures which mirrors the scientific discovery process. The learned structures can often decompose functions into interpretable components and enable long-range extrapolation on time-series datasets. Our structure search method outperforms many widely used kernels and kernel combination methods on a variety of prediction tasks.

David Lopez-Paz, José Miguel Hernández-Lobato, and Zoubin Ghahramani. Gaussian process vine copulas for multivariate dependence. In 30th International Conference on Machine Learning, Atlanta, Georgia, USA, June 2013.

Abstract: Copulas allow to learn marginal distributions separately from the multivariate dependence structure (copula) that links them together into a density function. Vine factorizations ease the learning of high-dimensional copulas by constructing a hierarchy of conditional bivariate copulas. However, to simplify inference, it is common to assume that each of these conditional bivariate copulas is independent from its conditioning variables. In this paper, we relax this assumption by discovering the latent functions that specify the shape of a conditional copula given its conditioning variables We learn these functions by following a Bayesian approach based on sparse Gaussian processes with expectation propagation for scalable, approximate inference. Experiments on real-world datasets show that, when modeling all conditional dependencies, we obtain better estimates of the underlying copula of the data.

Andrew Gordon Wilson and Ryan Prescott Adams. Gaussian process covariance kernels for pattern discovery and extrapolation. Technical Report arXiv:1302.4245 [stat.ML], Department of Engineering, University of Cambridge, Cambridge, UK, February 18 2013.

Abstract: Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns and enable extrapolation. These kernels are derived by modelling a spectral density - the Fourier transform of a kernel - with a Gaussian mixture. The proposed kernels support a broad class of stationary covariances, but Gaussian process inference remains simple and analytic. We demonstrate the proposed kernels by discovering patterns and performing long range extrapolation on synthetic examples, as well as atmospheric CO2 trends and airline passenger data. We also show that we can reconstruct standard covariances within our framework.

Comment: arXiv:1302.4245

Marc Peter Deisenroth, Dieter Fox, and Carl Edward Rasmussen. Gaussian processes for data-efficient learning in robotics and control. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013, doi 10.1109/TPAMI.2013.218.

Abstract: Autonomous learning has been a promising direction in control and robotics for more than a decade since data-driven learning allows to reduce the amount of engineering knowledge, which is otherwise required. However, autonomous reinforcement learning (RL) approaches typically require many interactions with the system to learn controllers, which is a practical limitation in real systems, such as robots, where many interactions can be impractical and time consuming. To address this problem, current learning approaches typically require task-speciﬁc knowledge in form of expert demonstrations, realistic simulators, pre-shaped policies, or speciﬁc knowledge about the underlying dynamics. In this article, we follow a different approach and speed up learning by extracting more information from data. In particular, we learn a probabilistic, non-parametric Gaussian process transition model of the system. By explicitly incorporating model uncertainty into long-term planning and controller learning our approach reduces the effects of model errors, a key problem in model-based learning. Compared to state-of-the art RL our model-based policy search method achieves an unprecedented speed of learning. We demonstrate its applicability to autonomous learning in real robot and control tasks.

Comment: accepted

Roger Frigola and Carl Edward Rasmussen. Integrated pre-processing for Bayesian nonlinear system identification with Gaussian processes. In Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, 2013.

Abstract: We introduce GP-FNARX: a new model for nonlinear system identification based on a nonlinear autoregressive exogenous model (NARX) with filtered regressors (F) where the nonlinear regression problem is tackled using sparse Gaussian processes (GP). We integrate data pre-processing with system identification into a fully automated procedure that goes from raw data to an identified model. Both pre-processing parameters and GP hyper-parameters are tuned by maximizing the marginal likelihood of the probabilistic model. We obtain a Bayesian model of the system's dynamics which is able to report its uncertainty in regions where the data is scarce. The automated approach, the modeling of uncertainty and its relatively low computational cost make of GP-FNARX a good candidate for applications in robotics and adaptive control.

E. Gilboa, Yunus Saatçi, and John P. Cunningham. Scaling multidimensional Gaussian processes using projected additive approximations. In 30th International Conference on Machine Learning, 2013.

Abstract: Exact Gaussian Process (GP) regression has O(N³) runtime for data size N, making it intractable for large N. Many algorithms for improving GP scaling approximate the covariance with lower rank matrices. Other work has exploited structure inherent in particular covariance functions, including GPs with implied Markov structure, and equispaced inputs (both enable O(N) runtime). However, these GP advances have not been extended to the multidimensional input setting, despite the preponderance of multidimensional applications. This paper introduces and tests novel extensions of structured GPs to multidimensional inputs. We present new methods for additive GPs, showing a novel connection between the classic backﬁtting method and the Bayesian framework. To achieve optimal accuracy-complexity tradeoff, we extend this model with a novel variant of projection pursuit regression. Our primary result – projection pursuit Gaussian Process Regression – shows orders of magnitude speedup while preserving high accuracy. The natural second and third steps include non-Gaussian observations and higher dimensional equispaced grid methods. We introduce novel techniques to address both of these necessary directions. We thoroughly illustrate the power of these three advances on several datasets, achieving close performance to the naive Full GP at orders of magnitude less cost.

E. Gilboa, Yunus Saatçi, and John P. Cunningham. Scaling multidimensional inference for structured Gaussian processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2013.

Comment: to appear

James Robert Lloyd. Gefcom2012 hierarchical load forecasting: Gradient boosting machines and gaussian processes. International Journal of Forecasting, 2013.

Abstract: This report discusses methods for forecasting hourly loads of a US utility as part of the load forecasting track of the Global Energy Forecasting Competition 2012 hosted on Kaggle. The methods described (gradient boosting machines and Gaussian processes) are generic machine learning / regression algorithms and few domain specific adjustments were made. Despite this, the algorithms were able to produce highly competitive predictions and hopefully they can inspire more reﬁned techniques to compete with state-of-the-art load forecasting methodologies.

David Duvenaud, Hannes Nickisch, and Carl Edward Rasmussen. Additive Gaussian processes. In Advances in Neural Information Processing Systems 24, pages 226-234, Granada, Spain, December 2012.

Abstract: We introduce a Gaussian process model of functions which are additive. An additive function is one which decomposes into a sum of low-dimensional functions, each depending on only a subset of the input variables. Additive GPs generalize both Generalized Additive Models, and the standard GP models which use squared-exponential kernels. Hyperparameter learning in this model can be seen as Bayesian Hierarchical Kernel Learning (HKL). We introduce an expressive but tractable parameterization of the kernel function, which allows efficient evaluation of all input interaction terms, whose number is exponential in the input dimension. The additional structure discoverable by this model results in increased interpretability, as well as state-of-the-art predictive power in regression tasks.

James Robert Lloyd, Peter Orbanz, Zoubin Ghahramani, and Daniel M. Roy. Random function priors for exchangeable arrays with applications to graphs and relational data. In Advances in Neural Information Processing Systems 26, pages 1-9, Lake Tahoe, California, USA, December 2012.

Abstract: A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common structure underlying relations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a flexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efficient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases.

Andrew McHutchon and Carl Edward Rasmussen. Gaussian process training with input noise. In Advances in Neural Information Processing Systems 24, Granada, Spain, December 2012.

Abstract: In standard Gaussian Process regression input locations are assumed to be noise free. We present a simple yet effective GP model for training on input points corrupted by i.i.d. Gaussian noise. To make computations tractable we use a local linear expansion about each input point. This allows the input noise to be recast as output noise proportional to the squared gradient of the GP posterior mean. The input noise hyperparameters are trained alongside other hyperparameters by the usual method of maximisation of the marginal likelihood, and allow estimation of the noise levels on each input dimension. Training uses an iterative scheme, which alternates between optimising the hyperparameters and calculating the posterior gradient. Analytic predictive moments can then be found for Gaussian distributed test points. We compare our model to others over a range of different regression problems and show that it improves over current methods.

Andrew Gordon Wilson, David A. Knowles, and Zoubin Ghahramani. Gaussian process regression networks. In 29th International Conference on Machine Learning, Edinburgh, Scotland, June 2012.

Abstract: We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the nonparametric flexibility of Gaussian processes. GPRN accommodates input (predictor) dependent signal and noise correlations between multiple output (response) variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both elliptical slice sampling and variational Bayes inference procedures for GPRN. We apply GPRN as a multiple output regression and multivariate volatility model, demonstrating substantially improved performance over eight popular multiple output (multi-task) Gaussian process models and three multivariate volatility models on real datasets, including a 1000 dimensional gene expression dataset.

Richard E. Turner and Maneesh Sahani. Decomposing signals into a sum of amplitude and frequency modulated sinusoids using probabilistic inference. In Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on, pages 2173-2176, march 2012, doi 10.1109/ICASSP.2012.6288343.

Abstract: There are many methods for decomposing signals into a sum of amplitude and frequency modulated sinusoids. In this paper we take a new estimation based approach. Identifying the problem as ill-posed, we show how to regularize the solution by imposing soft constraints on the amplitude and phase variables of the sinusoids. Estimation proceeds using a version of Kalman smoothing. We evaluate the method on synthetic and natural, clean and noisy signals, showing that it outperforms previous decompositions, but at a higher computational cost.

John P. Cunningham, Zoubin Ghahramani, and Carl E. Rasmussen. Gaussian Processes for time-marked time-series data. In 15th International Conference on Artificial Intelligence and Statistics, 2012.

Abstract: In many settings, data is collected as multiple time series, where each recorded time series is an observation of some underlying dynamical process of interest. These observations are often time-marked with known event times, and one desires to do a range of standard analyses. When there is only one time marker, one simply aligns the observations temporally on that marker. When multiple time-markers are present and are at different times on different time series observations, these analyses are more difficult. We describe a Gaussian Process model for analyzing multiple time series with multiple time markings, and we test it on a variety of data.

Marc Peter Deisenroth, Ryan D. Turner, Marco F. Huber, Uwe D. Hanebeck, and Carl Edward Rasmussen. Robust filtering and smoothing with Gaussian processes. IEEE Transactions on Automatic Control, 57(7):1865-1871, 2012.

Abstract: We propose a principled algorithm for robust Bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by nonparametric Gaussian process (GP) models. GPs are gaining increasing importance in signal processing, machine learning, robotics, and control for representing unknown system functions by posterior probability distributions. This modern way of "system identification" is more robust than finding point estimates of a parametric function representation. Our principled filtering/smoothing approach for GP dynamic systems is based on analytic moment matching in the context of the forward-backward algorithm. Our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state-of-the-art Gaussian filters and smoothers can fail.

Neil Houlsby, Jose Miguel Hernández-Lobato, Ferenc Huszár, and Zoubin Ghahramani. Collaborative Gaussian processes for preference learning. In Advances in Neural Information Processing Systems 26, pages 2096-2104. Curran Associates, Inc., 2012.

Abstract: We present a new model based on Gaussian processes (GPs) for learning pairwise preferences expressed by multiple users. Inference is simplified by using a preference kernel for GPs which allows us to combine supervised GP learning of user preferences with unsupervised dimensionality reduction for multi-user systems. The model not only exploits collaborative information from the shared structure in user behavior, but may also incorporate user features if they are available. Approximate inference is implemented using a combination of expectation propagation and variational Bayes. Finally, we present an efficient active learning strategy for querying preferences. The proposed technique performs favorably on real-world data against state-of-the-art multi-user preference learning algorithms.

Joseph Hall, Carl Edward Rasmussen, and Jan Maciejowski. Modelling and control of nonlinear systems using Gaussian processes with partial model information. In 51st IEEE Conference on Decision and Control, 2012.

Abstract: Gaussian processes are gaining increasing popularity among the control community, in particular for the modelling of discrete time state space systems. However, it has not been clear how to incorporate model information, in the form of known state relationships, when using a Gaussian process as a predictive model. An obvious example of known prior information is position and velocity related states. Incorporation of such information would be beneficial both computationally and for faster dynamics learning. This paper introduces a method of achieving this, yielding faster dynamics learning and a reduction in computational effort from O(Dn²) to O((D-F)n²) in the prediction stage for a system with D states, F known state relationships and n observations. The effectiveness of the method is demonstrated through its inclusion in the PILCO learning algorithm with application to the swing-up and balance of a torque-limited pendulum and the balancing of a robotic unicycle in simulation.

Ryan D. Turner and Carl Edward Rasmussen. Model based learning of sigma points in unscented Kalman filtering. Neurocomputing, 80:47-53, 2012, doi 10.1016/j.neucom.2011.07.029.

Abstract: The unscented Kalman filter (UKF) is a widely used method in control and time series applications. The UKF suffers from arbitrary parameters necessary for sigma point placement, potentially causing it to perform poorly in nonlinear problems. We show how to treat sigma point placement in a UKF as a learning problem in a model based view. We demonstrate that learning to place the sigma points correctly from data can make sigma point collapse much less likely. Learning can result in a significant increase in predictive performance over default settings of the parameters in the UKF and other filters designed to avoid the problems of the UKF, such as the GP-ADF. At the same time, we maintain a lower computational complexity than the other methods. We call our method UKF-L.

Andrew Gordon Wilson, David A Knowles, and Zoubin Ghahramani. Gaussian process regression networks. Technical Report arXiv:1110.4411 [stat.ML], Department of Engineering, University of Cambridge, Cambridge, UK, October 19 2011.

Abstract: We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian processes. This model accommodates input dependent signal and noise correlations between multiple response variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both efficient Markov chain Monte Carlo and variational Bayes inference procedures for this model. We apply GPRN as a multiple output regression and multivariate volatility model, demonstrating substantially improved performance over eight popular multiple output (multi-task) Gaussian process models and three multivariate volatility models on benchmark datasets, including a 1000 dimensional gene expression dataset.

Comment: arXiv:1110.4411

Simon Lacoste-Julien, Ferenc Huszár, and Zoubin Ghahramani. Approximate inference for the loss-calibrated Bayesian. In Geoff Gordon and David Dunson, editors, 14th International Conference on Artificial Intelligence and Statistics, volume 15, pages 416-424, Fort Lauderdale, FL, USA, April 2011. Journal of Machine Learning Research.

Abstract: We consider the problem of approximate inference in the context of Bayesian decision theory. Traditional approaches focus on approximating general properties of the posterior, ignoring the decision task - and associated losses - for which the posterior could be used. We argue that this can be suboptimal and propose instead to loss-calibrate the approximate inference methods with respect to the decision task at hand. We present a general framework rooted in Bayesian decision theory to analyze approximate inference from the perspective of losses, opening up several research directions. As a first loss-calibrated approximate inference attempt, we propose an EM-like algorithm on the Bayesian posterior risk and show how it can improve a standard approach to Gaussian process classification when losses are asymmetric.

Marc Peter Deisenroth and Carl Edward Rasmussen. PILCO: A model-based and data-efficient approach to policy search. In 28th International Conference on Machine Learning, 2011.

Abstract: In this paper, we introduce PILCO, a practical, data-efficient model-based policy search method. PILCO reduces model bias, one of the key problems of model-based reinforcement learning, in a principled way. By learning a probabilistic dynamics model and explicitly incorporating model uncertainty into long-term planning, PILCO can cope with very little data and facilitates learning from scratch in only a few trials. Policy evaluation is performed in closed form using state-of-the-art approximate inference. Furthermore, policy gradients are computed analytically for policy improvement. We report unprecedented learning efficiency on challenging and high-dimensional control tasks.

Comment: web site

Yunus Saatçi. Scalable inference for structured Gaussian process models. PhD thesis, University of Cambridge, Department of Engineering, Cambridge, UK, 2011.

Abstract: The generic inference and learning algorithm for Gaussian Process (GP) regression has O(N³) runtime and O(N²) memory complexity, where N is the number of observations in the dataset. Given the computational resources available to a present-day workstation, this implies that GP regression simply cannot be run on large datasets. The need to use non- Gaussian likelihood functions for tasks such as classification adds even more to the computational burden involved.
The majority of algorithms designed to improve the scaling of GPs are founded on the idea of approximating the true covariance matrix, which is usually of rank N, with a matrix of rank P, where P<<N. Typically, the true training set is replaced with a smaller, representative (pseudo-) training set such that a specific measure of information loss is minimized. These algorithms typically attain O(P²N) runtime and O(PN) space complexity. They are also general in the sense that they are designed to work with any covariance function. In essence, they trade off accuracy with computational complexity. The central contribution of this thesis is to improve scaling instead by exploiting any structure that is present in the covariance matrices generated by particular covariance functions. Instead of settling for a kernel-independent accuracy/complexity trade off, as is done in much the literature, we often obtain accuracies close to, or exactly equal to the full GP model at a fraction of the computational cost.
We define a structured GP as any GP model that is endowed with a kernel which produces structured covariance matrices. A trivial example of a structured GP is one with the linear regression kernel. In this case, given inputs living in R^D, the covariance matrices generated have rank D - this results in significant computational gains in the usual case where D<<N. Another case arises when a stationary kernel is evaluated on equispaced, scalar inputs. This results in Toeplitz covariance matrices and all necessary computations can be carried out exactly in O(N log N).
This thesis studies four more types of structured GP. First, we comprehensively review the case of kernels corresponding to Gauss-Markov processes evaluated on scalar inputs. Using state-space models we show how (generalised) regression (including hyperparameter learning) can be performed in O(N log N) runtime and O(N) space. Secondly, we study the case where we introduce block structure into the covariance matrix of a GP time-series model by assuming a particular form of nonstationarity a priori. Third, we extend the efficiency of scalar Gauss-Markov processes to higher-dimensional input spaces by assuming additivity. We illustrate the connections between the classical backfitting algorithm and approximate Bayesian inference techniques including Gibbs sampling and variational Bayes. We also show that it is possible to relax the rather strong assumption of additivity without sacrificing O(N log N) complexity, by means of a projection-pursuit style GP regression model. Finally, we study the properties of a GP model with a tensor product kernel evaluated on a multivariate grid of inputs locations. We show that for an arbitrary (regular or irregular) grid the resulting covariance matrices are Kronecker and full GP regression can be implemented in O(N) time and memory usage.
We illustrate the power of these methods on several real-world regression datasets which satisfy the assumptions inherent in the structured GP employed. In many cases we obtain performance comparable to the generic GP algorithm. We also analyse the performance degradation when these assumptions are not met, and in several cases show that it is comparable to that observed for sparse GP methods. We provide similar results for regression tasks with non-Gaussian likelihoods, an extension rarely addressed by sparse GP techniques.

Ryan Darby Turner. Gaussian processes for state space models and change point detection. PhD thesis, University of Cambridge, Department of Engineering, Cambridge, UK, 2011.

Abstract: This thesis details several applications of Gaussian processes (GPs) for enhanced time series modeling. We first cover different approaches for using Gaussian processes in time series problems. These are extended to the state space approach to time series in two different problems. We also combine Gaussian processes and Bayesian online change point detection (BOCPD) to increase the generality of the Gaussian process time series methods. These methodologies are evaluated on predictive performance on six real world data sets, which include three environmental data sets, one financial, one biological, and one from industrial well drilling.
Gaussian processes are capable of generalizing standard linear time series models. We cover two approaches: the Gaussian process time series model (GPTS) and the autoregressive Gaussian process (ARGP). We cover a variety of methods that greatly reduce the computational and memory complexity of Gaussian process approaches, which are generally cubic in computational complexity.
Two different improvements to state space based approaches are covered. First, Gaussian process inference and learning (GPIL) generalizes linear dynamical systems (LDS), for which the Kalman filter is based, to general nonlinear systems for nonparametric system identification. Second, we address pathologies in the unscented Kalman filter (UKF). We use Gaussian process optimization (GPO) to learn UKF settings that minimize the potential for sigma point collapse.
We show how to embed mentioned Gaussian process approaches to time series into a change point framework. Old data, from an old regime, that hinders predictive performance is automatically and elegantly phased out. The computational improvements for Gaussian process time series approaches are of even greater use in the change point framework. We also present a supervised framework learning a change point model when change point labels are available in training.
These mentioned methodologies significantly improve predictive performance on the diverse set of data sets selected.

Richard E. Turner and Maneesh Sahani. Demodulation as probabilistic inference. Transactions on Audio, Speech and Language Processing, 19:2398-2411, 2011.

Abstract: Demodulation is an ill-posed problem whenever both carrier and envelope signals are broadband and unknown. Here, we approach this problem using the methods of probabilistic inference. The new approach, called Probabilistic Amplitude Demodulation (PAD), is computationally challenging but improves on existing methods in a number of ways. By contrast to previous approaches to demodulation, it satisfies five key desiderata: PAD has soft constraints because it is probabilistic; PAD is able to automatically adjust to the signal because it learns parameters; PAD is user-steerable because the solution can be shaped by user-specific prior information; PAD is robust to broad-band noise because this is modelled explicitly; and PAD’s solution is self-consistent, empirically satisfying a Carrier Identity property. Furthermore, the probabilistic view naturally encompasses noise and uncertainty, allowing PAD to cope with missing data and return error bars on carrier and envelope estimates. Finally, we show that when PAD is applied to a bandpass-filtered signal, the stop-band energy of the inferred carrier is minimal, making PAD well-suited to sub-band demodulation.

Richard E. Turner and Maneesh Sahani. Probabilistic amplitude and frequency demodulation. In Advances in Neural Information Processing Systems 24, pages 981-989. The MIT Press, 2011.

Abstract: A number of recent scientific and engineering problems require signals to be decomposed into a product of a slowly varying positive envelope and a quickly varying carrier whose instantaneous frequency also varies slowly over time. Although signal processing provides algorithms for so-called amplitude- and frequency-demodulation (AFD), there are well known problems with all of the existing methods. Motivated by the fact that AFD is ill-posed, we approach the problem using probabilistic inference. The new approach, called probabilistic amplitude and frequency demodulation (PAFD), models instantaneous frequency using an auto-regressive generalization of the von Mises distribution, and the envelopes using Gaussian auto-regressive dynamics with a positivity constraint. A novel form of expectation propagation is used for inference. We demonstrate that although PAFD is computationally demanding, it outperforms previous approaches on synthetic and real signals in clean, noisy and missing data settings.

Andrew Gordon Wilson and Zoubin Ghahramani. Generalised Wishart processes. In 27nd Conference on Uncertainty in Artificial Intelligence, 2011.

Abstract: We introduce a new stochastic process called the generalised Wishart process (GWP). It is a collection of positive semi-definite random matrices indexed by any arbitrary input variable. We use this process as a prior over dynamic (e.g. time varying) covariance matrices. The GWP captures a diverse class of covariance dynamics, naturally hanles missing data, scales nicely with dimension, has easily interpretable parameters, and can use input variables that include covariates other than time. We describe how to construct the GWP, introduce general procedures for inference and prediction, and show that it outperforms its main competitor, multivariate GARCH, even on financial data that especially suits GARCH.

Comment: Supplementary Material, Best Student Paper Award

Carl Edward Rasmussen and Hannes Nickisch. Gaussian Processes for Machine Learning (GPML) Toolbox. Journal of Machine Learning Research, 11:3011-3015, December 2010.

Abstract: The GPML toolbox provides a wide range of functionality for Gaussian process (GP) inference and prediction. GPs are specified by mean and covariance functions; we offer a library of simple mean and covariance functions and mechanisms to compose more complex ones. Several likelihood functions are supported including Gaussian and heavy-tailed for regression as well as others suitable for classification. Finally, a range of inference methods is provided, including exact and variational inference, Expectation Propagation, and Laplace's method dealing with non-Gaussian likelihoods and FITC for dealing with large regression tasks.

Comment: Toolbox avaiable from here. Implements algorithms from Rasmussen and Williams, 2006.

Hannes Nickisch and Carl Edward Rasmussen. Gaussian mixture modeling with Gaussian process latent variable models. In Proceedings of the 32nd DAGM Symposium on Pattern Recognition, Lecture Notes in Computer Science (LNCS), Darmstadt, Germany, September 2010. Springer, doi 10.1007/978-3-642-15986-2_28.

Abstract: Density modeling is notoriously difficult for high dimensional data. One approach to the problem is to search for a lower dimensional manifold which captures the main characteristics of the data. Recently, the Gaussian Process Latent Variable Model (GPLVM) has successfully been used to find low dimensional manifolds in a variety of complex data. The GPLVM consists of a set of points in a low dimensional latent space, and a stochastic map to the observed space. We show how it can be interpreted as a density model in the observed space. However, the GPLVM is not trained as a density model and therefore yields bad density estimates. We propose a new training strategy and obtain improved generalisation performance and better density estimates in comparative evaluations on several benchmark data sets.

Ryan Turner and Carl Edward Rasmussen. Model based learning of sigma points in unscented Kalman filtering. In Samuel Kaski, David J. Miller, Erkki Oja, and Antti Honkela, editors, Machine Learning for Signal Processing (MLSP 2010), pages 178-183, Kittilä, Finland, August 2010.

Abstract: The unscented Kalman filter (UKF) is a widely used method in control and time series applications. The UKF suffers from arbitrary parameters necessary for a step known as sigma point placement, causing it to perform poorly in nonlinear problems. We show how to treat sigma point placement in a UKF as a learning problem in a model based view. We demonstrate that learning to place the sigma points correctly from data can make sigma point collapse much less likely. Learning can result in a significant increase in predictive performance over default settings of the parameters in the UKF and other filters designed to avoid the problems of the UKF, such as the GP-ADF. At the same time, we maintain a lower computational complexity than the other methods. We call our method UKF-L.

Miguel Lázaro-Gredilla, Joaquin Quiñonero-Candela, Carl Edward Rasmussen, and Aníbal Figueiras-Vidal. Sparse spectrum Gaussian process regression. Journal of Machine Learning Research, 11:1865-1881, June 2010.

Abstract: We present a new sparse Gaussian Process (GP) model for regression. The key novel idea is to sparsify the spectral representation of the GP. This leads to a simple, practical algorithm for regression tasks. We compare the achievable trade-offs between predictive accuracy and computational requirements, and show that these are typically superior to existing state-of-the-art sparse approximations. We discuss both the weight space and function space representations, and note that the new construction implies priors over functions which are always stationary, and can approximate any covariance function in this class.

Yunus Saatçi, Ryan Turner, and Carl Edward Rasmussen. Gaussian process change point models. In 27th International Conference on Machine Learning, pages 927-934, Haifa, Israel, June 2010.

Abstract: We combine Bayesian online change point detection with Gaussian processes to create a nonparametric time series model which can handle change points. The model can be used to locate change points in an online manner; and, unlike other Bayesian online change point detection algorithms, is applicable when temporal correlations in a regime are expected. We show three variations on how to apply Gaussian processes in the change point context, each with their own advantages. We present methods to reduce the computational burden of these models and demonstrate it on several real world data sets.

Comment: poster, slides.

Ryan Turner, Marc Peter Deisenroth, and Carl Edward Rasmussen. State-space inference and learning with Gaussian processes. In Yee Whye Teh and Mike Titterington, editors, 13th International Conference on Artificial Intelligence and Statistics, volume 9 of W & CP, pages 868-875, Chia Laguna, Sardinia, Italy, May 13-15 2010. Journal of Machine Learning Research.

Abstract: State-space inference and learning with Gaussian processes (GPs) is an unsolved problem. We propose a new, general methodology for inference and learning in nonlinear state-space models that are described probabilistically by non-parametric GP models. We apply the expectation maximization algorithm to iterate between inference in the latent state-space and learning the parameters of the underlying GP dynamics model.

Comment: poster.

M. M. Churchland, B. M. Yu, J. P. Cunningham, L. P. Sugrue, M. R. Cohen, G. S. Corrado, W. T. Newsome, A. M. Clark, P. Hosseini, B. B. Scott, D. C. Bradley, M. A. Smith, A. Kohn, J. A. Movshon, K. M. Armstrong, T. Moore, S. W. Chang, L. H. Snyder, S. G. Lisberger, N. J. Priebe, I. M. Finn, D. Ferster, S. I. Ryu, G. Santhanam, M. Sahani, and K. V. Shenoy. Stimulus onset quashes neural variability: a widespread cortical phenomenon. Nature Neuroscience, 13:369-378, 2010.

Abstract: Neural responses are typically characterized by computing the mean firing rate, but response variability can exist across trials. Many studies have examined the effect of a stimulus on the mean response, but few have examined the effect on response variability. We measured neural variability in 13 extracellularly recorded datasets and one intracellularly recorded dataset from seven areas spanning the four cortical lobes in monkeys and cats. In every case, stimulus onset caused a decline in neural variability. This occurred even when the stimulus produced little change in mean firing rate. The variability decline was observed in membrane potential recordings, in the spiking of individual neurons and in correlated spiking variability measured with implanted 96-electrode arrays. The variability decline was observed for all stimuli tested, regardless of whether the animal was awake, behaving or anaesthetized. This widespread variability decline suggests a rather general property of cortex, that its state is stabilized by an input.

Marc Peter Deisenroth. Efficient reinforcement learning using Gaussian processes. PhD thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2010.

Abstract: In many research areas, including control and medical applications, we face decision-making problems where data are limited and/or the underlying generative process is complicated and partially unknown. In these scenarios, we can profit from algorithms that learn from data and aid decision making.
Reinforcement learning (RL) is a general computational approach to experience-based goal-directed learning for sequential decision making under uncertainty. However, RL often lacks efficiency in terms of the number of required trials when no task-specific knowledge is available. This lack of efficiency makes RL often inapplicable to (optimal) control problems. Thus, a central issue in RL is to speed up learning by extracting more information from available experience.
The contributions of this dissertation are threefold:
1. We propose PILCO, a fully Bayesian approach for efficient RL in continuous-valued state and action spaces when no expert knowledge is available. PILCO is based on well-established ideas from statistics and machine learning. PILCO's key ingredient is a probabilistic dynamics model learned from data, which is implemented by a Gaussian process (GP). The GP carefully quantifies knowledge by a probability distribution over plausible dynamics models. By averaging over all these models during long-term planning and decision making, PILCO takes uncertainties into account in a principled way and, therefore, reduces model bias, a central problem in model-based RL.
2. Due to its generality and efficiency, PILCO can be considered a conceptual and practical approach to jointly learning models and controllers when expert knowledge is difficult to obtain or simply not available. For this scenario, we investigate PILCO's properties its applicability to challenging real and simulated nonlinear control problems. For example, we consider the tasks of learning to swing up a double pendulum attached to a cart or to balance a unicycle with five degrees of freedom. Across all tasks we report unprecedented automation and an unprecedented learning efficiency for solving these tasks.
3. As a step toward pilco's extension to partially observable Markov decision processes, we propose a principled algorithm for robust filtering and smoothing in GP dynamic systems. Unlike commonly used Gaussian filters for nonlinear systems, it does neither rely on function linearization nor on finite-sample representations of densities. Our algorithm profits from exact moment matching for predictions while keeping all computations analytically tractable. We present experimental evidence that demonstrates the robustness and the advantages of our method over unscented Kalman filters, the cubature Kalman filter, and the extended Kalman filter.

Richard E. Turner and Maneesh Sahani. Statistical inference for single- and multi-band probabilistic amplitude demodulation.. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pages 5466-5469, 2010.

Abstract: Amplitude demodulation is an ill-posed problem and so it is natural to treat it from a Bayesian viewpoint, inferring the most likely carrier and envelope under probabilistic constraints. One such treatment is Probabilistic Amplitude Demodulation (PAD), which, whilst computationally more intensive than traditional approaches, offers several advantages. Here we provide methods for estimating the uncertainty in the PAD-derived envelopes and carriers, and for learning free-parameters like the time-scale of the envelope. We show how the probabilistic approach can naturally handle noisy and missing data. Finally, we indicate how to extend the model to signals which contain multiple modulators and carriers.

Richard E. Turner. Statistical Models for Natural Sounds. PhD thesis, Gatsby Computational Neuroscience Unit, UCL, 2010.

Abstract: It is important to understand the rich structure of natural sounds in order to solve important tasks, like automatic speech recognition, and to understand auditory processing in the brain. This thesis takes a step in this direction by characterising the statistics of simple natural sounds. We focus on the statistics because perception often appears to depend on them, rather than on the raw waveform. For example the perception of auditory textures, like running water, wind, fire and rain, depends on summary-statistics, like the rate of falling rain droplets, rather than on the exact details of the physical source. In order to analyse the statistics of sounds accurately it is necessary to improve a number of traditional signal processing methods, including those for amplitude demodulation, time-frequency analysis, and sub-band demodulation. These estimation tasks are ill-posed and therefore it is natural to treat them as Bayesian inference problems. The new probabilistic versions of these methods have several advantages. For example, they perform more accurately on natural signals and are more robust to noise, they can also fill-in missing sections of data, and provide error-bars. Furthermore, free-parameters can be learned from the signal. Using these new algorithms we demonstrate that the energy, sparsity, modulation depth and modulation time-scale in each sub-band of a signal are critical statistics, together with the dependencies between the sub-band modulators. In order to validate this claim, a model containing co-modulated coloured noise carriers is shown to be capable of generating a range of realistic sounding auditory textures. Finally, we explored the connection between the statistics of natural sounds and perception. We demonstrate that inference in the model for auditory textures qualitatively replicates the primitive grouping rules that listeners use to understand simple acoustic scenes. This suggests that the auditory system is optimised for the statistics of natural sounds.

Andrew Gordon Wilson and Zoubin Ghahramani. Copula processes. In Advances in Neural Information Processing Systems 23, 2010. Spotlight.

Abstract: We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

Comment: Supplementary Material, slides.

Ryan Turner, Marc Peter Deisenroth, and Carl Edward Rasmussen. System identification in Gaussian process dynamical systems. In Dilan Görür, editor, NIPS Workshop on Nonparametric Bayes, Whistler, BC, Canada, December 2009.

Comment: poster.

B. M. Yu, J. P. Cunningham, G. Santhanam, S. I. Ryu, K. V. Shenoy, and M. Sahani. Gaussian-process factor analysis for low-dimensional single-trial analysis of neural population activity. In Advances in Neural Information Processing Systems 21, pages 1-8, Vancouver, BC, December 2009.

Abstract: We consider the problem of extracting smooth, low-dimensional neural trajectories that summarize the activity recorded simultaneously from many neurons on individual experimental trials. Beyond the benefit of visualizing the high-dimensional, noisy spiking activity in a compact form, such trajectories can offer insight into the dynamics of the neural circuitry underlying the recorded activity. Current methods for extracting neural trajectories involve a two-stage process: the spike trains are first smoothed over time, then a static dimensionality- reduction technique is applied. We first describe extensions of the two-stage methods that allow the degree of smoothing to be chosen in a principled way and that account for spiking variability, which may vary both across neurons and across time. We then present a novel method for extracting neural trajectories - Gaussian-process factor analysis (GPFA) - which unifies the smoothing and dimensionality- reduction operations in a common probabilistic framework. We applied these methods to the activity of 61 neurons recorded simultaneously in macaque premotor and motor cortices during reach planning and execution. By adopting a goodness-of-fit metric that measures how well the activity of each neuron can be predicted by all other recorded neurons, we found that the proposed extensions improved the predictive ability of the two-stage methods. The predictive ability was further improved by going to GPFA. From the extracted trajectories, we directly observed a convergence in neural state during motor planning, an effect that was shown indirectly by previous studies. We then show how such methods can be a powerful tool for relating the spiking activity across a neural population to the subject's behavior on a single-trial basis. Finally, to assess how well the proposed methods characterize neural population activity when the underlying time course is known, we performed simulations that revealed that GPFA performed tens of percent better than the best two-stage method.

R. Adams and Zoubin Ghahramani. Archipelago: nonparametric Bayesian semi-supervised learning. In Léon Bottou and Michael Littman, editors, 26th International Conference on Machine Learning, pages 1-8, Montréal, QC, Canada, June 2009. Omnipress.

Abstract: Semi-supervised learning (SSL), is classification where additional unlabeled data can be used to improve accuracy. Generative approaches are appealing in this situation, as a model of the data's probability density can assist in identifying clusters. Nonparametric Bayesian methods, while ideal in theory due to their principled motivations, have been difficult to apply to SSL in practice. We present a nonparametric Bayesian method that uses Gaussian processes for the generative model, avoiding many of the problems associated with Dirichlet process mixture models. Our model is fully generative and we take advantage of recent advances in Markov chain Monte Carlo algorithms to provide a practical inference method. Our method compares favorably to competing approaches on synthetic and real-world multi-class data.

Comment: This paper was awarded Honourable Mention for Best Paper at ICML 2009.

Marc Peter Deisenroth, Marco F. Huber, and Uwe D. Hanebeck. Analytic moment-based Gaussian process filtering. In Léon Bottou and Michael Littman, editors, 26th International Conference on Machine Learning, pages 225-232, Montréal, QC, Canada, June 2009. Omnipress.

Abstract: We propose an analytic moment-based filter for nonlinear stochastic dynamic systems modeled by Gaussian processes. Exact expressions for the expected value and the covariance matrix are provided for both the prediction step and the filter step, where an additional Gaussian assumption is exploited in the latter case. Our filter does not require further approximations. In particular, it avoids finite-sample approximations. We compare the filter to a variety of Gaussian filters, that is, the EKF, the UKF, and the recent GP-UKF proposed by Ko et al. (2007).

Comment: With corrections. code.

Marc Peter Deisenroth, Carl Edward Rasmussen, and Jan Peters. Gaussian process dynamic programming. Neurocomputing, 72(7-9):1508-1524, March 2009, doi 10.1016/j.neucom.2008.12.019.

Abstract: Reinforcement learning (RL) and optimal control of systems with continuous states and actions require approximation techniques in most interesting cases. In this article, we introduce Gaussian process dynamic programming (GPDP), an approximate value function-based RL algorithm. We consider both a classic optimal control problem, where problem-specific prior knowledge is available, and a classic RL problem, where only very general priors can be used. For the classic optimal control problem, GPDP models the unknown value functions with Gaussian processes and generalizes dynamic programming to continuous-valued states and actions. For the RL problem, GPDP starts from a given initial state and explores the state space using Bayesian active learning. To design a fast learner, available data have to be used efficiently. Hence, we propose to learn probabilistic models of the a priori unknown transition dynamics and the value functions on the fly. In both cases, we successfully apply the resulting continuous-valued controllers to the under-actuated pendulum swing up and analyze the performances of the suggested algorithms. It turns out that GPDP uses data very efficiently and can be applied to problems, where classic dynamic programming would be cumbersome.

Comment: code.

C. Chang, J. P. Cunningham, and G. Glover. Influence of heart rate on the bold signal: the cardiac response function. NeuroImage, 44:857-869, 2009.

Abstract: It has previously been shown that low-frequency fluctuations in both respiratory volume and cardiac rate can induce changes in the blood-oxygen level dependent (BOLD) signal. Such physiological noise can obscure the detection of neural activation using fMRI, and it is therefore important to model and remove the effects of this noise. While a hemodynamic response function relating respiratory variation (RV) and the BOLD signal has been described, no such mapping for heart rate (HR) has been proposed. In the current study, the effects of RV and HR are simultaneously deconvolved from resting state fMRI. It is demonstrated that a convolution model including RV and HR can explain significantly more variance in gray matter BOLD signal than a model that includes RV alone, and an average HR response function is proposed that well characterizes our subject population. It is observed that the voxel-wise morphology of the deconvolved RV responses is preserved when HR is included in the model, and that its form is adequately modeled by Birn et al.'s previously described respiration response function. Furthermore, it is shown that modeling out RV and HR can significantly alter functional connectivity maps of the default-mode network.

J. P. Cunningham, B. M. Yu, K. V. Shenoy, and M. Sahani. Inferring neural firing rates from spike trains using Gaussian processes. In Advances in Neural Information Processing Systems 20, pages 1-8, Vancouver, BC, December 2008.

Abstract: Neural spike trains present challenges to analytical efforts due to their noisy, spiking nature. Many studies of neuroscientific and neural prosthetic importance rely on a smoothed, denoised estimate of the spike train's underlying firing rate. Current techniques to find time-varying firing rates require ad hoc choices of parameters, offer no confidence intervals on their estimates, and can obscure potentially important single trial variability. We present a new method, based on a Gaussian Process prior, for inferring probabilistically optimal estimates of firing rate functions underlying single or multiple neural spike trains. We test the performance of the method on simulated data and experimentally gathered neural spike trains, and we demonstrate improvements over conventional estimators.

Comment: Spotlight Presentation

H. Kim and Zoubin Ghahramani. Outlier robust Gaussian process classification. In L. Niels da Vitoria, editor, Structural, Syntactic and Statistical Pattern Recognition, volume 5342 of Lecture Notes in Computer Science (LNCS), pages 896-905, Berlin, Germany, December 2008. Springer Berlin / Heidelberg.

Abstract: Gaussian process classifiers (GPCs) are a fully statistical model for kernel classification. We present a form of GPC which is robust to labeling errors in the data set. This model allows label noise not only near the class boundaries, but also far from the class boundaries which can result from mistakes in labelling or gross errors in measuring the input features. We derive an outlier robust algorithm for training this model which alternates iterations based on the EP approximation and hyperparameter updates until convergence. We show the usefulness of the proposed algorithm with model selection method through simulation results.

Hannes Nickisch and Carl Edward Rasmussen. Approximations for binary Gaussian process classification. Journal of Machine Learning Research, 9:2035-2078, October 2008.

Abstract: We provide a comprehensive overview of many recent algorithms for approximate inference in Gaussian process models for probabilistic binary classification. The relationships between several approaches are elucidated theoretically, and the properties of the different algorithms are corroborated by experimental results. We examine both 1) the quality of the predictive distributions and 2) the suitability of the different marginal likelihood approximations for model selection (selecting hyperparameters) and compare to a gold standard based on MCMC. Interestingly, some methods produce good predictive distributions although their marginal likelihood approximations are poor. Strong conclusions are drawn about the methods: The Expectation Propagation algorithm is almost always the method of choice unless the computational budget is very tight. We also extend existing methods in various ways, and provide unifying code implementing all approaches.

J. P. Cunningham, K. V. Shenoy, and M. Sahani. Fast Gaussian process methods for point process intensity estimation. In 25th International Conference on Machine Learning, pages 1-8, Helsinki, Finland, June 2008.

Abstract: Point processes are difficult to analyze because they provide only a sparse and noisy observation of the intensity function driving the process. Gaussian Processes offer an attractive framework within which to infer underlying intensity functions. The result of this inference is a continuous function defined across time that is typically more amenable to analytical efforts. However, a naive implementation will become computationally infeasible in any problem of reasonable size, both in memory and run time requirements. We demonstrate problem specific methods for a class of renewal processes that eliminate the memory burden and reduce the solve time by orders of magnitude.

Marc Peter Deisenroth, Jan Peters, and Carl Edward Rasmussen. Approximate dynamic programming with Gaussian processes. In 2008 American Control Conference (ACC 2008), pages 4480-4485, Seattle, WA, USA, June 2008.

Abstract: In general, it is difficult to determine an optimal closed-loop policy in nonlinear control problems with continuous-valued state and control domains. Hence, approximations are often inevitable. The standard method of discretizing states and controls suffers from the curse of dimensionality and strongly depends on the chosen temporal sampling rate. The paper introduces Gaussian Process Dynamic Programming (GPDP). In GPDP, value functions in the Bellman recursion of the dynamic programming algorithm are modeled using Gaussian processes. GPDP returns an optimal state-feedback for a finite set of states. Based on these outcomes, we learn a possibly discontinuous closed-loop policy on the entire state space by switching between two independently trained Gaussian processes.

Comment: code.

Marc Peter Deisenroth, Carl Edward Rasmussen, and Jan Peters. Model-based reinforcement learning with continuous states and actions. In Proceedings of the 16th European Symposium on Artificial Neural Networks (ESANN 2008), pages 19-24, Bruges, Belgium, April 2008.

Abstract: Finding an optimal policy in a reinforcement learning (RL) framework with continuous state and action spaces is challenging. Approximate solutions are often inevitable. GPDP is an approximate dynamic programming algorithm based on Gaussian process (GP) models for the value functions. In this paper, we extend GPDP to the case of unknown transition dynamics. After building a GP model for the transition dynamics, we apply GPDP to this model and determine a continuous-valued policy in the entire state space. We apply the resulting controller to the underpowered pendulum swing up. Moreover, we compare our results on this RL task to a nearly optimal discrete DP solution in a fully known environment.

Comment: code. slides

J. P. Cunningham. Derivation of Expectation Propagation for "fast Gaussian process methods for point process intensity estimation". Technical report, Stanford University, 2008.

Abstract: We derive the Expectation Propagation algorithm updates for approximating the posterior distribution on intensity in a conditionally inhomogeneous gamma interval process with a Gaussian Process prior (GP IGIP), a model which appeared in Cunningham, Shenoy, Sahani (2008) ICML.

Richard E. Turner and Maneesh Sahani. Modeling natural sounds with modulation cascade processes. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, nips20, volume 20. mit, 2008.

Abstract: Natural sounds are structured on many time-scales. A typical segment of speech, for example, contains features that span four orders of magnitude: Sentences ($\sim1$s); phonemes ($\sim10$−$1$ s); glottal pulses ($\sim 10$−$2$s); and formants ($\sim 10$−$3$s). The auditory system uses information from each of these time-scales to solve complicated tasks such as auditory scene analysis [1]. One route toward understanding how auditory processing accomplishes this analysis is to build neuroscience-inspired algorithms which solve similar tasks and to compare the properties of these algorithms with properties of auditory processing. There is however a discord: Current machine-audition algorithms largely concentrate on the shorter time-scale structures in sounds, and the longer structures are ignored. The reason for this is two-fold. Firstly, it is a difficult technical problem to construct an algorithm that utilises both sorts of information. Secondly, it is computationally demanding to simultaneously process data both at high resolution (to extract short temporal information) and for long duration (to extract long temporal information). The contribution of this work is to develop a new statistical model for natural sounds that captures structure across a wide range of time-scales, and to provide efficient learning and inference algorithms. We demonstrate the success of this approach on a missing data task.

W. Chu, V. Sindhwani, Z. Ghahramani, and S. Keerthi. Relational learning with Gaussian processes. In B. Schölkopf, J. Platt, and T. Hofmann, editors, Advances in Neural Information Processing Systems 19, volume 19 of Bradford Books, pages 289-296, Cambridge, MA, USA, September 2007. The MIT Press. Online contents gives pages 314-321, and 289-296 on pdf of contents.

Abstract: Correlation between instances is often modelled via a kernel function using input attributes of the instances. Relational knowledge can further reveal additional pairwise correlations between variables of interest. In this paper, we develop a class of models which incorporates both reciprocal relational information and input attributes using Gaussian process techniques. This approach provides a novel non-parametric Bayesian framework with a data-dependent prior for supervised learning tasks. We also apply this framework to semi-supervised learning. Experimental results on several real world data sets verify the usefulness of this algorithm.

Joaquin Quiñonero-Candela, Carl Edward Rasmussen, and Christopher K. I. Williams. Approximation methods for Gaussian process regression. In L. Bottou, O. Chapelle, D. DeCoste, and J. Weston, editors, Large-Scale Kernel Machines, Neural Information Processing, pages 203-223. The MIT Press, Cambridge, MA, USA, September 2007.

Abstract: A wealth of computationally efficient approximation methods for Gaussian process regression have been recently proposed. We give a unifying overview of sparse approximations, following Quiñonero-Candela and Rasmussen (2005), and a brief review of approximate matrix-vector multiplication methods.

Comment: book

Edward Snelson and Zoubin Ghahramani. Local and global sparse Gaussian process approximations. In M. Meila and X. Shen, editors, 11th International Conference on Artificial Intelligence and Statistics. Omnipress, 2007.

Abstract: Gaussian process (GP) models are flexible probabilistic nonparametric models for regression, classification and other tasks. Unfortunately they suffer from computational intractability for large data sets. Over the past decade there have been many different approximations developed to reduce this cost. Most of these can be termed global approximations, in that they try to summarize all the training data via a small set of support points. A different approach is that of local regression, where many local experts account for their own part of space. In this paper we start by investigating the regimes in which these different approaches work well or fail. We then proceed to develop a new sparse GP approximation which is a combination of both the global and local approaches. Theoretically we show that it is derived as a natural extension of the framework developed by Quiñonero-Candela and Rasmussen for sparse GP approximations. We demonstrate the benefits of the combined approximation on some 1D examples for illustration, and on some large real-world data sets.

Richard E. Turner and M Sahani. Probabilistic amplitude demodulation. In 7th International Conference on Independent Component Analysis and Signal Separation, pages 544-551, 2007.

Abstract: Auditory scene analysis is extremely challenging. One approach, perhaps that adopted by the brain, is to shape useful representations of sounds on prior knowledge about their statistical structure. For example, sounds with harmonic sections are common and so time-frequency representations are efficient. Most current representations concentrate on the shorter components. Here, we propose representations for structures on longer time-scales, like the phonemes and sentences of speech. We decompose a sound into a product of processes, each with its own characteristic time-scale. This demodulation cascade relates to classical amplitude demodulation, but traditional algorithms fail to realise the representation fully. A new approach, probabilistic amplitude demodulation, is shown to out-perform the established methods, and to easily extend to representation of a full demodulation cascade.

Malte Kuß and Carl Edward Rasmussen. Assessing approximations for Gaussian process classification. In Y. Weiss, B. Schölkopf, and J. Platt, editors, Advances in Neural Information Processing Systems 18, pages 699-706, Cambridge, MA, USA, April 2006. The MIT Press.

Abstract: Gaussian processes are attractive models for probabilistic classification but unfortunately exact inference is analytically intractable. We compare Laplace's method and Expectation Propagation (EP) focusing on marginal likelihood estimates and predictive performance. We explain theoretically and corroborate empirically that EP is superior to Laplace. We also compare to a sophisticated MCMC scheme and show that EP is surprisingly accurate.

Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. The MIT Press, 2006.

Abstract: Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics.

Comment: Winner of the 2009 DeGroot Prize. Book web page, chapters and entire book pdf. GPML Toolbox.

Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. In Y. Weiss, B. Schölkopf, and J. Platt, editors, Advances in Neural Information Processing Systems 18, pages 1257-1264. The MIT Press, Cambridge, MA, 2006.

Abstract: We present a new Gaussian process (GP) regression model whose covariance is parameterized by the the locations of M pseudo-input points, which we learn by a gradient based optimization. We take M<<N, where N is the number of real data points, and hence obtain a sparse regression method which has O(NM²) training cost and O(M²) prediction cost per test case. We also find hyperparameters of the covariance function in the same joint optimization. The method can be viewed as a Bayesian regression model with particular input dependent noise. The method turns out to be closely related to several other sparse GP approaches, and we discuss the relation in detail. We finally demonstrate its performance on some large data sets, and make a direct comparison to other sparse GP methods. We show that our method can match full GP performance with small M, i.e. very sparse solutions, and it significantly outperforms other approaches in this regime.

Edward Snelson and Zoubin Ghahramani. Variable noise and dimensionality reduction for sparse Gaussian processes. In R. Dechter and T. S. Richardson, editors, 22nd Conference on Uncertainty in Artificial Intelligence. AUAI Press, 2006.

Abstract: The sparse pseudo-input Gaussian process (SPGP) is a new approximation method for speeding up GP regression in the case of a large number of data points N. The approximation is controlled by the gradient optimization of a small set of M pseudo-inputs, thereby reducing complexity from O(N³) to O(NM²). One limitation of the SPGP is that this optimization space becomes impractically big for high dimensional data sets. This paper addresses this limitation by performing automatic dimensionality reduction. A projection of the input space to a low dimensional space is learned in a supervised manner, alongside the pseudo-inputs, which now live in this reduced space. The paper also investigates the suitability of the SPGP for modeling data with input-dependent noise. A further extension of the model is made to make it even more powerful in this regard - we learn an uncertainty parameter for each pseudo-input. The combination of sparsity, reduced dimension, and input-dependent noise makes it possible to apply GPs to much larger and more complex data sets than was previously practical. We demonstrate the benefits of these methods on several synthetic and real world problems.

Malte Kuß, Tobias Pfingsten, Lehel Csatò, and Carl Edward Rasmussen. Approximate inference for robust Gaussian process regression. Technical Report 136, Max Planck Institute for Biological Cybernetics, Tübingen, Germany, 2005.

Abstract: Gaussian process (GP) priors have been successfully used in non-parametric Bayesian regression and classification models. Inference can be performed analytically only for the regression model with Gaussian noise. For all other likelihood models inference is intractable and various approximation techniques have been proposed. In recent years expectation-propagation (EP) has been developed as a general method for approximate inference. This article provides a general summary of how expectation-propagation can be used for approximate inference in Gaussian process models. Furthermore we present a case study describing its implementation for a new robust variant of Gaussian process regression. To gain further insights into the quality of the EP approximation we present experiments in which we compare to results obtained by Markov chain Monte Carlo (MCMC) sampling.

Malte Kuß and Carl Edward Rasmussen. Assessing approximate inference for binary Gaussian process classification. Journal of Machine Learning Research, 6:1679-1704, 2005.

Abstract: Gaussian process priors can be used to define flexible, probabilistic classification models. Unfortunately exact Bayesian inference is analytically intractable and various approximation techniques have been proposed. In this work we review and compare Laplace's method and Expectation Propagation for approximate Bayesian inference in the binary Gaussian process classification model. We present a comprehensive comparison of the approximations, their predictive performance and marginal likelihood estimates to results obtained by MCMC sampling. We explain theoretically and corroborate empirically the advantages of Expectation Propagation compared to Laplace's method.

Joaquin Quiñonero-Candela and Carl Edward Rasmussen. Analysis of some methods for reduced rank Gaussian process regression. In R. Murray-Smith and R. Shorten, editors, Switching and Learning in Feedback Systems, pages 98-127. Springer, Berlin, Heidelberg, 2005.

Abstract: While there is strong motivation for using Gaussian Processes (GPs) due to their excellent performance in regression and classification problems, their computational complexity makes them impractical when the size of the training set exceeds a few thousand cases. This has motivated the recent proliferation of a number of cost-effective approximations to GPs, both for classification and for regression. In this paper we analyze one popular approximation to GPs for regression: the reduced rank approximation. While generally GPs are equivalent to infinite linear models, we show that Reduced Rank Gaussian Processes (RRGPs) are equivalent to finite sparse linear models. We also introduce the concept of degenerate GPs and show that they correspond to inappropriate priors. We show how to modify the RRGP to prevent it from being degenerate at test time. Training RRGPs consists both in learning the covariance function hyperparameters and the support set. We propose a method for learning hyperparameters for a given support set. We also review the Sparse Greedy GP (SGGP) approximation (Somla and Bartlett, 2001), which is a way of learning the support set for given hyperparameters based on approximating the posterior. We propose an alternative method to the SGGP that has better generalization capabilities. Finally we make experiments to compare the different ways of training a RRGP. We provide some Matlab code for learning RRGPs.

Joaquin Quiñonero-Candela and Carl Edward Rasmussen. A unifying view of sparse approximate Gaussian process regression. Journal of Machine Learning Research, 6:1939-1959, 2005.

Abstract: We provide a new unifying view, including all existing proper probabilistic sparse approximations for Gaussian process regression. Our approach relies on expressing the effective prior which the methods are using. This allows new insights to be gained, and highlights the relationship between existing methods. It also allows for a clear theoretically justified ranking of the closeness of the known approximations to the corresponding full GPs. Finally we point directly to designs of new better sparse approximations, combining the best of the existing strategies, within attractive computational constraints.

Carl Edward Rasmussen and Joaquin Quiñonero-Candela. Healing the Relevance Vector Machine through augmentation. In L. De Raedt and S. Wrobel, editors, 22nd International Conference on Machine Learning, pages 689-696, 2005.

Abstract: The Relevance Vector Machine (RVM) is a sparse approximate Bayesian kernel method. It provides full predictive distributions for test cases. However, the predictive uncertainties have the unintuitive property, that they get smaller the further you move away from the training cases. We give a thorough analysis. Inspired by the analogy to non-degenerate Gaussian Processes, we suggest augmentation to solve the problem. The purpose of the resulting model, RVM*, is primarily to corroborate the theoretical and experimental analysis. Although RVM* could be used in practical applications, it is no longer a truly sparse model. Experiments show that sparsity comes at the expense of worse predictive distributions.

Edward Snelson, Carl Edward Rasmussen, and Zoubin Ghahramani. Warped Gaussian processes. In S. Thrun, L. Saul, and B. Schölkopf, editors, Advances in Neural Information Processing Systems 16, pages 337-344, Cambridge, MA, USA, December 2004. The MIT Press.

Abstract: We generalise the Gaussian process (GP) framework for regression by learning a nonlinear transformation of the GP outputs. This allows for non-Gaussian processes and non-Gaussian noise. The learning algorithm chooses a nonlinear transformation such that transformed data is well-modelled by a GP. This can be seen as including a preprocessing transformation as an integral part of the probabilistic modelling problem, rather than as an ad-hoc step. We demonstrate on several real regression problems that learning the transformation can lead to significantly better performance than using a regular GP, or a GP with a fixed transformation.

Juš Kocijan, Roderick Murray-Smith, Carl Edward Rasmussen, and Agathe Girard. Gaussian process model based predictive control. In American Control Conference, pages 2214-2219, 2004.

Abstract: Gaussian process models provide a probabilistic non-parametric modelling approach for black-box identi cation of non-linear dynamic systems. The Gaussian processes can highlight areas of the input space where prediction quality is poor, due to the lack of data or its complexity, by indicating the higher variance around the predicted mean. Gaussian process models contain noticeably less coef cients to be optimised. This paper illustrates possible application of Gaussian process models within model-based predictive control. The extra information provided within Gaussian process model is used in predictive control, where optimisation of control signal takes the variance information into account. The predictive control principle is demonstrated on control of pH process benchmark.

Carl Edward Rasmussen. Gaussian processes in machine learning. In Olivier Bousquet, Ulrike von Luxburg, and Gunnar Rätsch, editors, Advanced Lectures on Machine Learning: ML Summer Schools 2003, Canberra, Australia, February 2 - 14, 2003, Tübingen, Germany, August 4 - 16, 2003, Revised Lectures, volume 3176 of Lecture Notes in Computer Science (LNCS), pages 63-71. Springer-Verlag, Heidelberg, 2004.

Abstract: We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with conclusions and a look at the current trends in GP work.

Comment: Copyright by Springer, springerlink

Fabian Sinz, Joaquin Quiñonero-Candela, Gökhan H. Bakir, Carl Edward Rasmussen, and Matthias O. Franz. Learning depth from stereo. In C. E. Rasmussen, H. H. Bülthoff, B. Schölkopf, and M. A. Giese, editors, 26th DAGM Symposium, volume 3175 of Lecture Notes in Computer Science (LNCS), pages 245-252, Berlin, Germany, 09 2004. Springer.

Abstract: We compare two approaches to the problem of estimating the depth of a point in space from observing its image position in two different cameras: 1. The classical photogrammetric approach explicitly models the two cameras and estimates their intrinsic and extrinsic parameters using a tedious calibration procedure; 2. A generic machine learning approach where the mapping from image to spatial coordinates is directly approximated by a Gaussian Process regression. Our results show that the generic learning approach, in addition to simplifying the procedure of calibration, can lead to higher depth accuracies than classical calibration although no specific domain knowledge is used.

Agathe Girard, Carl Edward Rasmussen, Joaquin Quiñonero-Candela, and Roderick Murray-Smith. Gaussian process priors with uncertain inputs - application to multiple-step ahead time series forecasting. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 529-536, Cambridge, MA, USA, December 2003. The MIT Press.

Abstract: We consider the problem of multi-step ahead prediction in time series analysis using the non-parametric Gaussian process model. k-step ahead forecasting of a discrete-time non-linear dynamic system can be performed by doing repeated one-step ahead predictions. For a state-space model of the form y_t = f(y_t-1,...,y_t-L), the prediction of y at time t + k is based on the point estimates of the previous outputs. In this paper, we show how, using an analytical Gaussian approximation, we can formally incorporate the uncertainty about intermediate regressor values, thus updating the uncertainty on the current prediction.

Carl Edward Rasmussen and Zoubin Ghahramani. Bayesian Monte Carlo. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 489-496, Cambridge, MA, USA, December 2003. The MIT Press.

Abstract: We investigate Bayesian alternatives to classical Monte Carlo methods for evaluating integrals. Bayesian Monte Carlo (BMC) allows the incorporation of prior knowledge, such as smoothness of the integrand, into the estimation. In a simple problem we show that this outperforms any classical importance sampling method. We also attempt more challenging multidimensional integrals involved in computing marginal likelihoods of statistical models (a.k.a. partition functions and model evidences). We find that Bayesian Monte Carlo outperformed Annealed Importance Sampling, although for very high dimensional problems or problems with massive multimodality BMC may be less adequate. One advantage of the Bayesian approach to Monte Carlo is that samples can be drawn from any distribution. This allows for the possibility of active design of sample points so as to maximise information gain.

Ercan Solak, Roderick Murray-Smith, William E. Leithead, Douglas Leith, and Carl Edward Rasmussen. Derivative observations in Gaussian process models of dynamic systems. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 1033-1040, Cambridge, MA, USA, December 2003. The MIT Press.

Abstract: Gaussian processes provide an approach to nonparametric modelling which allows a straightforward combination of function and derivative observations in an empirical model. This is of particular importance in identification of nonlinear dynamic systems from experimental data. 1) It allows us to combine derivative information, and associated uncertainty with normal function observations into the learning and inference process. This derivative information can be in the form of priors specified by an expert or identified from perturbation data close to equilibrium. 2) It allows a seamless fusion of multiple local linear models in a consistent manner, inferring consistent models and ensuring that integrability constraints are met. 3) It improves dramatically the computational efficiency of Gaussian process models for dynamic system identification, by summarising large quantities of near-equilibrium data by a handful of linearisations, reducing the training set size - traditionally a problem for Gaussian process models.

Roderick Murray-Smith, Daniel Sbarbaro, Carl Edward Rasmussen, and Agathe Girard. Adaptive, cautious, predictive control with Gaussian process priors. In P. Van den Hof, B. Wahlberg, and S. Weiland, editors, IFAC SYSID 2003, pages 1195-1200, Oxford, UK, August 2003. Elsevier Science Ltd.

Abstract: Nonparametric Gaussian Process models, a Bayesian statistics approach, are used to implement a nonlinear adaptive control law. Predictions, including propagation of the state uncertainty are made over a k-step horizon. The expected value of a quadratic cost function is minimised, over this prediction horizon, without ignoring the variance of the model predictions. The general method and its main features are illustrated on a simulation example.

Joaquin Quiñonero-Candela, Agathe Girard, Jan Larsen, and Carl Edward Rasmussen. Propagation of uncertainty in Bayesian kernel models - application to multiple-step ahead forecasting. In ICASSP 2003, volume 2, pages 701-704, April 2003.

Abstract: The object of Bayesian modelling is the predictive distribution, which in a forecasting scenario enables improved estimates of forecasted values and their uncertainties. In this paper we focus on reliably estimating the predictive mean and variance of forecasted values using Bayesian kernel based models such as the Gaussian Process and the Relevance Vector Machine. We derive novel analytic expressions for the predictive mean and variance for Gaussian kernel shapes under the assumption of a Gaussian input distribution in the static case, and of a recursive Gaussian predictive density in iterative forecasting. The capability of the method is demonstrated for forecasting of time-series and compared to approximate methods.

Juš Kocijan, Blaž Banko, Bojan Likar, Agathe Girard, Roderick Murray-Smith, and Carl Edward Rasmussen. A case based comparison of identification with neural network and Gaussian process models. In IFAC Internaltional Conference on Intelligent Control Systems and Signal Processing, volume 1, pages 137-142, 2003.

Abstract: In this paper an alternative approach to black-box identification of non-linear dynamic systems is compared with the more established approach of using artificial neural networks. The Gaussian process prior approach is a representative of non-parametric modelling approaches. It was compared on a pH process modelling case study. The purpose of modelling was to use the model for control design. The comparison revealed that even though Gaussian process models can be effectively used for modelling dynamic systems caution has to be axercised when signals are selected.

Juš Kocijan, Roderick Murray-Smith, Carl Edward Rasmussen, and Bojan Likar. Predictive control with Gaussian process models. In B. Zajc and M. Tkal, editors, IEEE Region 8 Eurocon 2003: Computer as a Tool, pages 352-356, 2003.

Abstract: This paper describes model-based predictive control based on Gaussian processes. Gaussian process models provide a probabilistic non-parametric modelling approach for black-box identification of non-linear dynamic systems. It offers more insight in variance of obtained model response, as well as fewer parameters to determine than other models. The Gaussian processes can highlight areas of the input space where prediction quality is poor, due to the lack of data or its complexity, by indicating the higher variance around the predicted mean. This property is used in predictive control, where optimisation of control signal takes the variance information into account. The predictive control principle is demonstrated on a simulated example of nonlinear system.

Joaquin Quiñonero-Candela, Agathe Girard, and Carl Edward Rasmussen. Prediction at an uncertain input for Gaussian processes and Relevance Vector Machines application to multiple-step ahead time-series prediction. Technical Report IMM-2003-18, Instititute for Mathemetical Modelling, DTU, 2003.

Comment: techreport

Carl Edward Rasmussen. Gaussian processes to speed up Hybrid Monte Carlo for expensive Bayesian integrals. In Bayesian Statistics 7, pages 651-659. Oxford University Press, 2003.

Abstract: Hybrid Monte Carlo (HMC) is often the method of choice for computing Bayesian integrals that are not analytically tractable. However the success of this method may require a very large number of evaluations of the (un-normalized) posterior and its partial derivatives. In situations where the posterior is computationally costly to evaluate, this may lead to an unacceptable computational load for HMC. I propose to use a Gaussian Process model of the (log of the) posterior for most of the computations required by HMC. Within this scheme only occasional evaluation of the actual posterior is required to guarantee that the samples generated have exactly the desired distribution, even if the GP model is somewhat inaccurate. The method is demonstrated on a 10 dimensional problem, where 200 evaluations suffice for the generation of 100 roughly independent points from the posterior. Thus, the proposed scheme allows Bayesian treatment of models with posteriors that are computationally demanding, such as models involving computer simulation.

Carl Edward Rasmussen and Zoubin Ghahramani. Infinite mixtures of Gaussian process experts. In Advances in Neural Information Processing Systems 14, pages 881-888, Cambridge, MA, USA, December 2002. The MIT Press.

Abstract: We present an extension to the Mixture of Experts (ME) model, where the individual experts are Gaussian Process (GP) regression models. Using an input-dependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this model may be done efficiently using a Markov Chain relying on Gibbs sampling. The model allows the effective covariance function to vary with the inputs, and may handle large datasets - thus potentially overcoming two of the biggest hurdles with GP models. Simulations show the viability of this approach.

Irene K. Andersen, Anna Szymkowiak, Carl Edward Rasmussen, L. G. Hanson, J. R. Marstrand, H. B. W. Larsson, and Lars Kai Hansen. Perfusion quantification using Gaussian process deconvolution. Magnetic Resonance in Medicine, 48(2):351-361, 2002, doi 10.1002/mrm.10213.

Abstract: The quantification of perfusion using dynamic susceptibility contrast MR imaging requires deconvolution to obtain the residual impulse-response function (IRF). Here, a method using a Gaussian process for deconvolution, GPD, is proposed. The fact that the IRF is smooth is incorporated as a constraint in the method. The GPD method, which automatically estimates the noise level in each voxel, has the advantage that model parameters are optimized automatically. The GPD is compared to singular value decomposition (SVD) using a common threshold for the singular values and to SVD using a threshold optimized according to the noise level in each voxel. The comparison is carried out using artificial data as well as using data from healthy volunteers. It is shown that GPD is comparable to SVD variable optimized threshold when determining the maximum of the IRF, which is directly related to the perfusion. GPD provides a better estimate of the entire IRF. As the signal to noise ratio increases or the time resolution of the measurements increases, GPD is shown to be superior to SVD. This is also found for large distribution volumes.

Christopher K. I. Williams, Carl Edward Rasmussen, Anton Schwaighofer, and Volker Tresp. Observations on the Nyström method for Gaussian process prediction. Technical report, University of Edinburgh, 2002.

Abstract: A number of methods for speeding up Gaussian Process (GP) prediction have been proposed, including the Nyström method of Williams and Seeger (2001). In this paper we focus on two issues (1) the relationship of the Nyström method to the Subset of Regressors method (Poggio and Girosi 1990; Luo and Wahba, 1997) and (2) understanding in what circumstances the Nyström approximation would be expected to provide a good approximation to exact GP regression.

Carl Edward Rasmussen. Evaluation of Gaussian processes and other methods for non-linear regression. PhD thesis, University of Toronto, Department of Computer Science, Toronto, CANADA, 1996.

Abstract: This thesis develops two Bayesian learning methods relying on Gaussian processes and a rigorous statistical approach for evaluating such methods. In these experimental designs the sources of uncertainty in the estimated generalisation performances due to both variation in training and test sets are accounted for. The framework allows for estimation of generalisation performance as well as statistical tests of significance for pairwise comparisons. Two experimental designs are recommended and supported by the DELVE software environment.
Two new non-parametric Bayesian learning methods relying on Gaussian process priors over functions are developed. These priors are controlled by hyperparameters which set the characteristic length scale for each input dimension. In the simplest method, these parameters are fit from the data using optimization. In the second, fully Bayesian method, a Markov chain Monte Carlo technique is used to integrate over the hyperparameters. One advantage of these Gaussian process methods is that the priors and hyperparameters of the trained models are easy to interpret.
The Gaussian process methods are benchmarked against several other methods, on regression tasks using both real data and data generated from realistic simulations. The experiments show that small datasets are unsuitable for benchmarking purposes because the uncertainties in performance measurements are large. A second set of experiments provide strong evidence that the bagging procedure is advantageous for the Multivariate Adaptive Regression Splines (MARS) method.
The simulated datasets have controlled characteristics which make them useful for understanding the relationship between properties of the dataset and the performance of different methods. The dependency of the performance on available computation time is also investigated. It is shown that a Bayesian approach to learning in multi-layer perceptron neural networks achieves better performance than the commonly used early stopping procedure, even for reasonably short amounts of computation time. The Gaussian process methods are shown to consistently outperform the more conventional methods.

Chris K. I. Williams and Carl Edward Rasmussen. Gaussian processes for regression. In Advances in Neural Information Processing Systems 8, pages 514-520, Cambridge, MA., USA, 1996. The MIT Press.

Abstract: The Bayesian analysis of neural networks is difficult because a simple prior over weights implies a complex prior over functions. We investigate the use of a Gaussian process prior over functions, which permits the predictive Bayesian analysis for fixed values of hyperparameters to be carried out exactly using matrix operations. Two methods, using optimization and averaging (via Hybrid Monte Carlo) over hyperparameters have been tested on a number of challenging problems and have produced excellent results.

Clustering

Clustering algorithms are unsupervised methods for finding groups of similar points in data. They are closely related to statistical mixture models.

P. Kirk, J. E. Griffin, R. S. Savage, Z. Ghahramani, and D. L. Wild. Bayesian correlated clustering to integrate multiple datasets. Bioinformatics, 2012.

Abstract: Motivation: The integration of multiple datasets remains a key challenge in systems biology and genomic medicine. Modern high-throughput technologies generate a broad array of different data types, providing distinct – but often complementary – information. We present a Bayesian method for the unsupervised integrative modelling of multiple datasets, which we refer to as MDI (Multiple Dataset Integration). MDI can integrate information from a wide range of different datasets and data types simultaneously (including the ability to model time series data explicitly using Gaussian processes). Each dataset is modelled using a Dirichlet-multinomial allocation (DMA) mixture model, with dependencies between these models captured via parameters that describe the agreement among the datasets.
Results: Using a set of 6 artificially constructed time series datasets, we show that MDI is able to integrate a significant number of datasets simultaneously, and that it successfully captures the underlying structural similarity between the datasets. We also analyse a variety of real S. cerevisiae datasets. In the 2-dataset case, we show that MDI’s performance is comparable to the present state of the art. We then move beyond the capabilities of current approaches and integrate gene expression, ChIP-chip and protein-protein interaction data, to identify a set of protein complexes for which genes are co-regulated during the cell cycle. Comparisons to other unsupervised data integration techniques – as well as to non-integrative approaches – demonstrate that MDI is very competitive, while also providing information that would be difficult or impossible to extract using other methods.

Comment: This paper is available from the Bioinformatics site and a Matlab implementation of MDI is available fromthis site.

Donglin Niu, Jennifer G. Dy, and Z. Ghahramani. A nonparametric Bayesian model for multiple clustering with overlapping feature views. In 15th International Conference on Artificial Intelligence and Statistics, 2012.

Abstract: Most clustering algorithms produce a single clustering solution. This is inadequate for many data sets that are multi-faceted and can be grouped and interpreted in many different ways. Moreover, for high-dimensional data, different features may be relevant or irrelevant to each clustering solution, suggesting the need for feature selection in clustering. Features relevant to one clustering interpretation may be different from the ones relevant for an alternative interpretation or view of the data. In this paper, we introduce a probabilistic nonparametric Bayesian model that can discover multiple clustering solutions from data and the feature subsets that are relevant for the clusters in each view. In our model, the features in different views may be shared and therefore the sets of relevant features are allowed to overlap. We model feature relevance to each view using an Indian Buffet Process and the cluster membership in each view using a Chinese Restaurant Process. We provide an inference approach to learn the latent parameters corresponding to this multiple partitioning problem. Our model not only learns the features and clusters in each view but also automatically learns the number of clusters, number of views and number of features in each view.

Joshua Abbott, Katherine A. Heller, Zoubin Ghahramani, and Thomas L. Griffiths. Testing a Bayesian measure of representativeness using a large image database. In Advances in Neural Information Processing Systems 24, Cambridge, MA, USA, 2011. The MIT Press.

Abstract: How do people determine which elements of a set are most representative of that set? We extend an existing Bayesian measure of representativeness, which indicates the representativeness of a sample from a distribution, to deﬁne a measure of the representativeness of an item to a set. We show that this measure is formally related to a machine learning method known as Bayesian Sets. Building on this connection, we derive an analytic expression for the representativeness of objects described by a sparse vector of binary features. We then apply this measure to a large database of images, using it to determine which images are the most representative members of different sets. Comparing the resulting predictions to human judgments of representativeness provides a test of this measure with naturalistic stimuli, and illustrates how databases that are more commonly used in computer vision and machine learning can be used to evaluate psychological theories.

Daniel Hernández-Lobato, José Miguel Hernández-Lobato, and Pierre Dupont. Robust multi-class Gaussian process classification. In Advances in Neural Information Processing Systems 25, 2011.

Abstract: Multi-class Gaussian Processs Classifiers (MGPCs) are often affected by overfitting problems when labeling errors occur far from the decision boundaries. To prevent this, we investigate a robust MGPC (RMGPC) which considers labeling errors independently of their distance to the decision boundaries. Expectation propagation is used for approximate inference. Experiments with several datasets in which noise is injected in the labels illustrate the benefits of RMGPC. This method performs better than other Gaussian process alternatives based on considering latent Gaussian noise or heavy-tailed processes. When no noise is injected in the labels, RMGPC still performs equal or better than the other methods. Finally, we show how RMGPC can be used for successfully indentifying data instances which are difficult to classify correctly in practice.

David A. Knowles and Zoubin Ghahramani. Pitman-Yor diffusion trees. In 27nd Conference on Uncertainty in Artificial Intelligence, 2011.

Abstract: We introduce the Pitman Yor Diffusion Tree (PYDT) for hierarchical clustering, a generalization of the Dirichlet Diffusion Tree (Neal, 2001) which removes the restriction to binary branching structure. The generative process is described and shown to result in an exchangeable distribution over data points. We prove some theoretical properties of the model and then present two inference methods: a collapsed MCMC sampler which allows us to model uncertainty over tree structures, and a computationally efficient greedy Bayesian EM search algorithm. Both algorithms use message passing on the tree structure. The utility of the model and algorithms is demonstrated on synthetic and real world data, both continuous and binary.

Comment: web site

David A. Knowles and Thomas P. Minka. Non-conjugate variational message passing for multinomial and binary regression. In Advances in Neural Information Processing Systems 25, 2011.

Abstract: Variational Message Passing (VMP) is an algorithmic implementation of the Variational Bayes (VB) method which applies only in the special case of conjugate exponential family models. We propose an extension to VMP, which we refer to as Non-conjugate Variational Message Passing (NCVMP) which aims to alleviate this restriction while maintaining modularity, allowing choice in how expectations are calculated, and integrating into an existing message-passing framework: Infer.NET. We demonstrate NCVMP on logistic binary and multinomial regression. In the multinomial case we introduce a novel variational bound for the softmax factor which is tighter than other commonly used bounds whilst maintaining computational tractability.

Comment: web site supplementary

Y. Guan, J. G. Dy, D. Niu, and Z. Ghahramani. Variational inference for nonparametric multiple clustering. In KDD10 Workshop on Discovering, Summarizing, and Using Multiple Clusterings, Washington, DC, USA, July 2010.

Abstract: Most clustering algorithms produce a single clustering solution. Similarly, feature selection for clustering tries to find one feature subset where one interesting clustering solution resides. However, a single data set may be multi-faceted and can be grouped and interpreted in many different ways, especially for high dimensional data, where feature selection is typically needed. Moreover, different clustering solutions are interesting for different purposes. Instead of committing to one clustering solution, in this paper we introduce a probabilistic nonparametric Bayesian model that can discover several possible clustering solutions and the feature subset views that generated each cluster partitioning simultaneously. We provide a variational inference approach to learn the features and clustering partitions in each view. Our model allows us not only to learn the multiple clusterings and views but also allows us to automatically learn the number of views and the number of clusters in each view.

R. P. Adams, Zoubin Ghahramani, and Michael I. Jordan. Tree-structured stick breaking for hierarchical data. In Advances in Neural Information Processing Systems 23. The MIT Press, 2010.

Abstract: Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.

Andreas Vlachos, Zoubin Ghahramani, and Ted Briscoe. Active learning for constrained Dirichlet process mixture models. In Proceedings of the 2010 Workshop on Geometrical Models of Natural Language Semantics, pages 57-61, Uppsala, Sweden, 2010.

Abstract: Recent work applied Dirichlet Process Mixture Models to the task of verb clustering, incorporating supervision in the form of must-links and cannot-links constraints between instances. In this work, we introduce an active learning approach for constraint selection employing uncertainty-based sampling. We achieve substantial improvements over random selection on two datasets.

R. Savage, K. A. Heller, Y. Xu, Zoubin Ghahramani, W. Truman, M. Grant, K. Denby, and D. L. Wild. R/BHC: fast Bayesian hierarchical clustering for microarray data. BMC Bioinformatics 2009, 10(242):1-9, August 2009, doi 10.1186/1471-2105-10-242.

Abstract: Background: Although the use of clustering methods has rapidly become one of the standard computational approaches in the literature of microarray gene expression data analysis, little attention has been paid to uncertainty in the results obtained.
Results: We present an R/Bioconductor port of a fast novel algorithm for Bayesian agglomerative hierarchical clustering and demonstrate its use in clustering gene expression microarray data. The method performs bottom-up hierarchical clustering, using a Dirichlet Process (infinite mixture) to model uncertainty in the data and Bayesian model selection to decide at each step which clusters to merge.
Conclusion: Biologically plausible results are presented from a well studied data set: expression profiles of A. thaliana subjected to a variety of biotic and abiotic stresses. Our method avoids several limitations of traditional methods, for example how many clusters there should be and how to choose a principled distance metric.

A. Vlachos, A Korhonen, and Z. Ghahramani. Unsupervised and constrained Dirichlet process mixture models for verb clustering. In 4th Workshop on Statistical Machine Translation, EACL '09, Athens, Greece, March 2009.

Abstract: In this work, we apply Dirichlet Process Mixture Models (DPMMs) to a learning task in natural language processing (NLP): lexical-semantic verb clustering. We thoroughly evaluate a method of guiding DPMMs towards a particular clustering solution using pairwise constraints. The quantitative and qualitative evaluation performed highlights the benefits of both standard and constrained DPMMs compared to previously used approaches. In addition, it sheds light on the use of evaluation measures and their practical application.

Carl Edward Rasmussen, Bernhard J. de la Cruz, Zoubin Ghahramani, and David L. Wild. Modeling and visualizing uncertainty in gene expression clusters using Dirichlet process mixtures. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 6(4):615-628, 2009, doi 10.1109/TCBB.2007.70269.

Abstract: Although the use of clustering methods has rapidly become one of the standard computational approaches in the literature of microarray gene expression data, little attention has been paid to uncertainty in the results obtained. Dirichlet process mixture (DPM) models provide a nonparametric Bayesian alternative to the bootstrap approach to modeling uncertainty in gene expression clustering. Most previously published applications of Bayesian model-based clustering methods have been to short time series data. In this paper, we present a case study of the application of nonparametric Bayesian clustering methods to the clustering of high-dimensional nontime series gene expression data using full Gaussian covariances. We use the probability that two genes belong to the same cluster in a DPM model as a measure of the similarity of these gene expression profiles. Conversely, this probability can be used to define a dissimilarity measure, which, for the purposes of visualization, can be input to one of the standard linkage algorithms used for hierarchical clustering. Biologically plausible results are obtained from the Rosetta compendium of expression profiles which extend previously published cluster analyses of this data.

Katherine A. Heller, Sinead Williamson, and Zoubin Ghahramani. Statistical models for partial membership. In Andrew McCallum and Sam Roweis, editors, 25th International Conference on Machine Learning, pages 392-399, Helsinki, Finland, July 2008. Omnipress.

Abstract: We present a principled Bayesian framework for modeling partial memberships of data points to clusters. Unlike a standard mixture model which assumes that each data point belongs to one and only one mixture component, or cluster, a partial membership model allows data points to have fractional membership in multiple clusters. Algorithms which assign data points partial memberships to clusters can be useful for tasks such as clustering genes based on microarray data (Gasch & Eisen, 2002). Our Bayesian Partial Membership Model (BPM) uses exponential family distributions to model each cluster, and a product of these distibtutions, with weighted parameters, to model each datapoint. Here the weights correspond to the degree to which the datapoint belongs to each cluster. All parameters in the BPM are continuous, so we can use Hybrid Monte Carlo to perform inference and learning. We discuss relationships between the BPM and Latent Dirichlet Allocation, Mixed Membership models, Exponential Family PCA, and fuzzy clustering. Lastly, we show some experimental results and discuss nonparametric extensions to our model.

A. Vlachos, Z. Ghahramani, and A Korhonen. Dirichlet process mixture models for verb clustering. In Guillaume Bouchard, Hal Daumé III, Marc Dymetman, and Yee Whye Teh, editors, ICML Workshop on Prior Knowledge for Text and Language Processing, pages 43-48, Helsinki, Finland, July 2008.

Abstract: In this work we apply Dirichlet Process Mixture Models to a learning task in natural language processing (NLP): lexical-semantic verb clustering. We assess the performance on a dataset based on Levin's (1993) verb classes using the recently introduced V-measure metric. In, we present a method to add human supervision to the model in order to to influence the solution with respect to some prior knowledge. The quantitative evaluation performed highlights the benefits of the chosen method compared to previously used clustering approaches.

Zoubin Ghahramani and Katherine A. Heller. Bayesian sets. In Y. Weiss, B. Schölkopf, and J. Platt, editors, Advances in Neural Information Processing Systems 18, pages 435-442, Cambridge, MA, USA, December 2006. The MIT Press.

Abstract: Inspired by "Google™ Sets", we consider the problem of retrieving items from a concept or cluster, given a query consisting of a few items from that cluster. We formulate this as a Bayesian inference problem and describe a very simple algorithm for solving it. Our algorithm uses a model-based concept of a cluster and ranks items using a score which evaluates the marginal probability that each item belongs to a cluster containing the query items. For exponential family models with conjugate priors this marginal probability is a simple function of sufficient statistics. We focus on sparse binary data and show that our score can be evaluated exactly using a single sparse matrix multiplication, making it possible to apply our algorithm to very large datasets. We evaluate our algorithm on three datasets: retrieving movies from EachMovie, finding completions of author sets from the NIPS dataset, and finding completions of sets of words appearing in the Grolier encyclopedia. We compare to Google™ Sets and show that Bayesian Sets gives very reasonable set completions.

A. Dubey, S. Hwang, C. Rangel, Carl Edward Rasmussen, Zoubin Ghahramani, and David L. Wild. Clustering protein sequence and structure space with infinite Gaussian mixture models. In Pacific Symposium on Biocomputing 2004, pages 399-410, Singapore, 2004. World Scientific Publishing.

Abstract: We describe a novel approach to the problem of automatically clustering protein sequences and discovering protein families, subfamilies etc., based on the thoery of infinite Gaussian mixture models. This method allows the data itself to dictate how many mixture components are required to model it, and provides a measure of the probability that two proteins belong to the same cluster. We illustrate our methods with application to three data sets: globin sequences, globin sequences with known tree-dimensional structures and G-pretein coupled receptor sequences. The consistency of the clusters indicate that that our methods is producing biologically meaningful results, which provide a very good indication of the underlying families and subfamilies. With the inclusion of secondary structure and residue solvent accessibility information, we obtain a classification of sequences of known structure which reflects and extends their SCOP classifications.

Graphical Models

Graphical models are a graphical representation of the conditional independence relations among a set of variables. The graph is useful both as an intuitive representation of how the variables are related, and as a tool for defining efficient message passing algorithms for probabilistic inference.

Daniel Hernández-Lobato and José Miguel Hernández-Lobato. Learning feature selection dependencies in multi-task learning. In Advances in Neural Information Processing Systems 27, pages 1-9, Lake Tahoe, California, USA, December 2013.

Abstract: A probabilistic model based on the horseshoe prior is proposed for learning de- pendencies in the process of identifying relevant features for prediction. Exact inference is intractable in this model. However, expectation propagation offers an approximate alternative. Because the process of estimating feature selection dependencies may suffer from over-fitting in the model proposed, additional data from a multi-task learning scenario are considered for induction. The same model can be used in this setting with few modifications. Furthermore, the assumptions made are less restrictive than in other multi-task methods: The different tasks must share feature selection dependencies, but can have different relevant features and model coefficients. Experiments with real and synthetic data show that this model performs better than other multi-task alternatives from the literature. The experiments also show that the model is able to induce suitable feature selection dependencies for the problems considered, only from the training data.

Daniel Hernández-Lobato, José Miguel Hernández-Lobato, and Pierre Dupont. Generalized spike-and-slab priors for Bayesian group feature selection using expectation propagation. Journal of Machine Learning Research, 14:1891-1945, July 2013.

Abstract: We describe a Bayesian method for group feature selection in linear regression problems. The method is based on a generalized version of the standard spike-and-slab prior distribution which is often used for individual feature selection. Exact Bayesian inference under the prior considered is infeasible for typical regression problems. However, approximate inference can be carried out efficiently using Expectation Propagation (EP). A detailed analysis of the generalized spike-and-slab prior shows that it is well suited for regression problems that are sparse at the group level. Furthermore, this prior can be used to introduce prior knowledge about specific groups of features that are a priori believed to be more relevant. An experimental evaluation compares the performance of the proposed method with those of group LASSO, Bayesian group LASSO, automatic relevance determination and additional variants used for group feature selection. The results of these experiments show that a model based on the generalized spike-and-slab prior and the EP algorithm has state-of-the-art prediction performance in the problems analyzed. Furthermore, this model is also very useful to carry out sequential experimental design (also known as active learning), where the data instances that are most informative are iteratively included in the training set, reducing the number of instances needed to obtain a particular level of prediction accuracy.

Yarin Gal and Phil Blunsom. A systematic Bayesian treatment of the IBM alignment models. In Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. Association for Computational Linguistics, 2013.

Abstract: The dominant yet ageing IBM and HMM word alignment models underpin most popular Statistical Machine Translation implementations in use today. Though beset by the limitations of implausible independence assumptions, intractable optimisation problems, and an excess of tunable parameters, these models provide a scalable and reliable starting point for inducing translation systems. In this paper we build upon this venerable base by recasting these models in the non-parametric Bayesian framework. By replacing the categorical distributions at their core with hierarchical Pitman-Yor processes, and through the use of collapsed Gibbs sampling, we provide a more flexible formulation and sidestep the original heuristic optimisation techniques. The resulting models are highly extendible, naturally permitting the introduction of phrasal dependencies. We present extensive experimental results showing improvements in both AER and BLEU when benchmarked against Giza++, including significant improvements over IBM model 4.

José Miguel Hernández-Lobato, Neil Houlsby, and Zoubin Ghahramani. Stochastic inference for scalable probabilistic modeling of binary matrices. In NIPS Workshop on Randomized Methods for Machine Learning, 2013.

Abstract: Fully observed large binary matrices appear in a wide variety of contexts. To model them, probabilistic matrix factorization (PMF) methods are an attractive solution. However, current batch algorithms for PMF can be inefficient since they need to analyze the entire data matrix before producing any parameter updates. We derive an efficient stochastic inference algorithm for PMF models of fully observed binary matrices. Our method exhibits faster convergence rates than more expensive batch approaches and has better predictive performance than scalable alternatives. The proposed method includes new data subsampling strategies which produce large gains over standard uniform subsampling. We also address the task of automatically selecting the size of the minibatches of data and we propose an algorithm that adjusts this hyper-parameter in an online manner.

Daniel M. Roy. On the computability and complexity of Bayesian reasoning. In NIPS Workshop on Philosophy and Machine Learning, 2011.

Abstract: If we consider the claim made by some cognitive scientists that the mind performs Bayesian reasoning, and if we simultaneously accept the Physical Church-Turing thesis and thus believe that the computational power of the mind is no more than that of a Turing machine, then what limitations are there to the reasoning abilities of the mind? I give an overview of joint work with Nathanael Ackerman (Harvard, Mathematics) and Cameron Freer (MIT, CSAIL) that bears on the computability and complexity of Bayesian reasoning. In particular, we prove that conditional probability is in general not computable in the presence of continuous random variables. However, in light of additional structure in the prior distribution, such as the presence of certain types of noise, or of exchangeability, conditioning is possible. These results cover most of statistical practice. At the workshop on Logic and Computational Complexity, we presented results on the computational complexity of conditioning, embedding sharp-P-complete problems in the task of computing conditional probabilities for diffuse continuous random variables. This work complements older work. For example, under cryptographic assumptions, the computational complexity of producing samples and computing probabilities was separated by Ben-David, Chor, Goldreich and Luby. In recent work, we also make use of cryptographic assumptions to show that different representations of exchangeable sequences may have vastly different complexity. However, when faced with an adversary that is computational bounded, these different representations have the same complexity, highlighting the fact that knowledge representation and approximation play a fundamental role in the possibility and plausibility of Bayesian reasoning.

R. P. Adams, H. Wallach, and Zoubin Ghahramani. Learning the structure of deep sparse graphical models. In Yee Whye Teh and Mike Titterington, editors, 13th International Conference on Artificial Intelligence and Statistics, pages 1-8, Chia Laguna, Sardinia, Italy, May 2010.

Abstract: Deep belief networks are a powerful way to model complex probability distributions. However, it is difficult to learn the structure of a belief network, particularly one with hidden units. The Indian buffet process has been used as a nonparametric Bayesian prior on the structure of a directed belief network with a single infinitely wide hidden layer. Here, we introduce the cascading Indian buffet process (CIBP), which provides a prior on the structure of a layered, directed belief network that is unbounded in both depth and width, yet allows tractable inference. We use the CIBP prior with the nonlinear Gaussian belief network framework to allow each unit to vary its behavior between discrete and continuous representations. We use Markov chain Monte Carlo for inference in this model and explore the structures learned on image data.

Comment: Winner of the Best Paper Award

Charalampos Rotsos, Jurgen Van Gael, Andrew W. Moore, and Zoubin Ghahramani. Probabilistic graphical models for semi-supervised traffic classification. In The 6th International Wireless Communications and Mobile Computing Conference, pages 752-757, Caen, France, 2010.

Abstract: Traffic classification using machine learning continues to be an active research area. The majority of work in this area uses off-the-shelf machine learning tools and treats them as black-box classifiers. This approach turns all the modelling complexity into a feature selection problem. In this paper, we build a problem-specific solution to the traffic classification problem by designing a custom probabilistic graphical model. Graphical models are a modular framework to design classifiers which incorporate domain-specific knowledge. More specifically, our solution introduces semi-supervised learning which means we learn from both labelled and unlabelled traffic flows. We show that our solution performs competitively compared to previous approaches while using less data and simpler features.

R. Silva and Z. Ghahramani. The hidden life of latent variables: Bayesian learning with mixed graph models. Journal of Machine Learning Research, 10:1187-1238, June 2009.

Abstract: Directed acyclic graphs (DAGs) have been widely used as a representation of conditional independence in machine learning and statistics. Moreover, hidden or latent variables are often an important component of graphical models. However, DAG models suffer from an important limitation: the family of DAGs is not closed under marginalization of hidden variables. This means that in general we cannot use a DAG to represent the independencies over a subset of variables in a larger DAG. Directed mixed graphs (DMGs) are a representation that includes DAGs as a special case, and overcomes this limitation. This paper introduces algorithms for performing Bayesian inference in Gaussian and probit DMG models. An important requirement for inference is the specification of the distribution over parameters of the models. We introduce a new distribution for covariance matrices of Gaussian DMGs. We discuss and illustrate how several Bayesian machine learning tasks can benefit from the principle presented here: the power to model dependencies that are generated from hidden variables, but without necessarily modeling such variables explicitly.

Frederik Eaton and Zoubin Ghahramani. Choosing a variable to clamp: Approximate inference using conditioned belief propagation. In D. van Dyk and M. Welling, editors, 12th International Conference on Artificial Intelligence and Statistics, volume 5, pages 145-152, Clearwater Beach, FL, USA, April 2009. Journal of Machine Learning Research.

Abstract: In this paper we propose an algorithm for approximate inference on graphical models based on belief propagation (BP). Our algorithm is an approximate version of Cutset Conditioning, in which a subset of variables is instantiated to make the rest of the graph singly connected. We relax the constraint of single-connectedness, and select variables one at a time for conditioning, running belief propagation after each selection. We consider the problem of determining the best variable to clamp at each level of recursion, and propose a fast heuristic which applies back-propagation to the BP updates. We demonstrate that the heuristic performs better than selecting variables at random, and give experimental results which show that it performs competitively with existing approximate inference algorithms.

Comment: Code (in C++ based on libDAI).

R. Silva and Z. Ghahramani. Factorial mixture of Gaussians and the marginal independence model. In 12th International Conference on Artificial Intelligence and Statistics, volume 5, pages 520-527, Clearwater Beach, FL, USA, April 2009. Journal of Machine Learning Research. ISSN: 1938-7228.

Abstract: Marginal independence constraints play an important role in learning with graphical models. One way of parameterizing a model of marginal independencies is by building a latent variable model where two independent observed variables have no common latent source. In sparse domains, however, it might be advantageous to model the marginal observed distribution directly, without explicitly including latent variables in the model. There have been recent advances in Gaussian and binary models of marginal independence, but no models with non-linear dependencies between continuous variables has been proposed so far. In this paper, we describe how to generalize the Gaussian model of marginal independencies based on mixtures, and how to learn parameters. This requires a non-standard parameterization and raises difficult non-linear optimization issues.

Comment: Code at http://www.homepages.ucl.ac.uk/~ucgtrbd/code/fmog-version0.zip

R. Silva, W. Chu, and Zoubin Ghahramani. Hidden common cause relations in relational learning. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1345-1352, Cambridge, MA, USA, December 2008. The MIT Press.

Abstract: When predicting class labels for objects within a relational database, it is often helpful to consider a model for relationships: this allows for information between class labels to be shared and to improve prediction performance. However, there are different ways by which objects can be related within a relational database. One traditional way corresponds to a Markov network structure: each existing relation is represented by an undirected edge. This encodes that, conditioned on input features, each object label is independent of other object labels given its neighbors in the graph. However, there is no reason why Markov networks should be the only representation of choice for symmetric dependence structures. Here we discuss the case when relationships are postulated to exist due to hidden common causes. We discuss how the resulting graphical model differs from Markov networks, and how it describes different types of real-world relational processes. A Bayesian nonparametric classification model is built upon this graphical representation and evaluated with several empirical studies.

Comment: Code at http://www.homepages.ucl.ac.uk/~ucgtrbd/code/xgp

J. Zhang, Z. Ghahramani, and Y. Yang. Flexible latent variable models for multi-task learning. Machine Learning, 73(3):221-242, December 2008.

Abstract: Given multiple prediction problems such as regression and classification, we are interested in a joint inference framework which can effectively borrow information among tasks to improve the prediction accuracy, especially when the number of training examples per problem is small. In this paper we propose a probabilistic framework which can support a set of latent variable models for different multi-task learning scenarios. We show that the framework is a generalization of standard learning methods for single prediction problems and it can effectively model the shared structure among different prediction tasks. Furthermore, we present efficient algorithms for the empirical Bayes method as well as point estimation. Our experiments on both simulated datasets and real world classification datasets show the effectiveness of the proposed models in two evaluation settings: standard multi-task learning setting and transfer learning setting.

F. Pérez-Cruz, Zoubin Ghahramani, and M. Pontil. Conditional graphical models. In G. H. Bakir, T. Hofmann, B. Schölkopf, A. J. Smola, B. Taskar, and S. V. N. Vishwanathan, editors, Predicting Structured Data, pages 265-282. The MIT Press, Cambridge, MA, USA, September 2007. Chapter 12.

Abstract: In this chapter we propose a modification of CRF-like algorithms that allows for solving large-scale structured classification problems. Our approach consists in upper bounding the CRF functional in order to decompose its training into independent optimisation problems per clique. Furthermore we show that each sub-problem corresponds to solving a multiclass learning task in each clique, which enlarges the applicability of these tools for large-scale structural learning problems. Before presenting the Conditional Graphical Model (CGM), as we refer to this procedure, we review the family of CRF algorithms. We concentrate on the best known procedures and standard generalisations of CRFs. The ob jective of this introduction is analysing from the same viewpoint the proposed solutions in the literature to tackle this problem, which allows comparing their different features. We complete the chapter with a case study, in which we show the possibility to work with large-scale problems using CGM and that the obtained performance is comparable to the result with CRF-like algorithms.

Monte Carlo Methods

Markov chain Monte Carlo (MCMC) methods use sampling to approximate high dimensional integrals and intractable sums. MCMC methods are widely used in many areas of science, applied mathematics and engineering. They are an indispensable approximate inference tool for Bayesian statistics and machine learning.

Yue Wu, José Miguel Hernández-Lobato, and Zoubin Ghahramani. Dynamic covariance models for multivariate financial time series. In 30th International Conference on Machine Learning, Atlanta, Georgia, USA, June 2013.

Abstract: The accurate prediction of time-changing covariances is an important problem in the modeling of multivariate financial data. However, some of the most popular models suffer from a) overfitting problems and multiple local optima, b) failure to capture shifts in market conditions and c) large computational costs. To address these problems we introduce a novel dynamic model for time-changing covariances. Over-fitting and local optima are avoided by following a Bayesian approach instead of computing point estimates. Changes in market conditions are captured by assuming a diffusion process in parameter values, and finally computationally efficient and scalable inference is performed using particle filters. Experiments with financial data show excellent performance of the proposed method with respect to current standard models.

Sébastien Bratières, Jurgen Van Gael, Andreas Vlachos, and Zoubin Ghahramani. Scaling the iHMM: Parallelization versus Hadoop. In Proceedings of the 2010 10th IEEE International Conference on Computer and Information Technology, pages 1235-1240, Bradford, UK, 2010. IEEE Computer Society, doi 10.1109/CIT.2010.223.

Abstract: This paper compares parallel and distributed implementations of an iterative, Gibbs sampling, machine learning algorithm. Distributed implementations run under Hadoop on facility computing clouds. The probabilistic model under study is the infinite HMM Beal, Ghahramani and Rasmussen, 2002, in which parameters are learnt using an instance blocked Gibbs sampling, with a step consisting of a dynamic program. We apply this model to learn part-of-speech tags from newswire text in an unsupervised fashion. However our focus here is on runtime performance, as opposed to NLP-relevant scores, embodied by iteration duration, ease of development, deployment and debugging.

Finale Doshi-Velez and Zoubin Ghahramani. Accelerated Gibbs sampling for the Indian buffet process. In Léon Bottou and Michael Littman, editors, 26th International Conference on Machine Learning, pages 273-280, Montréal, QC, Canada, June 2009. Omnipress.

Abstract: We often seek to identify co-occurring hidden features in a set of observations. The Indian Buffet Process (IBP) provides a non-parametric prior on the features present in each observation, but current inference techniques for the IBP often scale poorly. The collapsed Gibbs sampler for the IBP has a running time cubic in the number of observations, and the uncollapsed Gibbs sampler, while linear, is often slow to mix. We present a new linear-time collapsed Gibbs sampler for conjugate likelihood models and demonstrate its efficacy on large real-world datasets.

C. Hübler, K. Borgwardt, H.-P. Kriegel, and Z. Ghahramani. Metropolis algorithms for representative subgraph sampling. In Proceedings of 8th IEEE International Conference on Data Mining (ICDM 2008), pages 283-292, Pisa, Italy, December 2008. IEEE. ISSN: 1550-4786.

Abstract: While data mining in chemoinformatics studied graph data with dozens of nodes, systems biology and the Internet are now generating graph data with thousands and millions of nodes. Hence data mining faces the algorithmic challenge of coping with this significant increase in graph size: Classic algorithms for data analysis are often too expensive and too slow on large graphs.
While one strategy to overcome this problem is to design novel efficient algorithms, the other is to 'reduce' the size of the large graph by sampling. This is the scope of this paper: We will present novel Metropolis algorithms for sampling a 'representative' small subgraph from the original large graph, with 'representative' describing the requirement that the sample shall preserve crucial graph properties of the original graph. In our experiments, we improve over the pioneering work of Leskovec and Faloutsos (KDD 2006), by producing representative subgraph samples that are both smaller and of higher quality than those produced by other methods from the literature.

Jurgen Van Gael, Yunus Saatçi, Yee-Whye Teh, and Zoubin Ghahramani. Beam sampling for the infinite hidden Markov model. In 25th International Conference on Machine Learning, volume 25, pages 1088-1095, Helsinki, Finland, 2008. ACM.

Abstract: The infinite hidden Markov model is a non-parametric extension of the widely used hidden Markov model. Our paper introduces a new inference algorithm for the infinite Hidden Markov model called beam sampling. Beam sampling combines slice sampling, which limits the number of states considered at each time step to a finite number, with dynamic programming, which samples whole state trajectories efficiently. Our algorithm typically outperforms the Gibbs sampler and is more robust. We present applications of iHMM inference using the beam sampler on changepoint detection and text prediction problems.

Carl Edward Rasmussen. Gaussian processes to speed up Hybrid Monte Carlo for expensive Bayesian integrals. In Bayesian Statistics 7, pages 651-659. Oxford University Press, 2003.

Carl Edward Rasmussen. A practical Monte Carlo implementation of Bayesian learning. In Advances in Neural Information Processing Systems 8, pages 598-604, Cambridge, MA., USA, 1996. The MIT Press.

Abstract: A practical method for Bayesian training of feed-forward neural networks using sophisticated Monte Carlo methods is presented and evaluated. In reasonably small amounts of computer time this approach outperforms other state-of-the-art methods on 5 datalimited tasks from real world domains.

Semi-Supervised Learning

Often, it is easy and cheap to obtain large amounts of unlabelled data (e.g. images, text documents), while it is hard or expensive to obtain labelled data. Semi-supervised learning methods attempt to use the unlabelled data to improve the performance on supervised learning tasks, such as classification.

David Lopez-Paz, José Miguel Hernández-Lobato, and Bernhard Scholköpf. Semi-supervised domain adaptation with non-parametric copulas. In Advances in Neural Information Processing Systems 26, pages 1-9, Lake Tahoe, California, USA, December 2012.

Abstract: A new framework based on the theory of copulas is proposed to address semisupervised domain adaptation problems. The presented method factorizes any multivariate density into a product of marginal distributions and bivariate copula functions. Therefore, changes in each of these factors can be detected and corrected to adapt a density model accross different learning domains. Importantly, we introduce a novel vine copula model, which allows for this factorization in a non-parametric manner. Experimental results on regression problems with real-world data illustrate the efficacy of the proposed approach when compared to state-of-the-art techniques.

C. Rotsos, J. Van Gael, A.W. Moore, and Z. Ghahramani. Traffic classification in information poor environments. In 1st International Workshop on Traffic Analysis and Classification (IWCMC '10), Caen, France, July 2010.

Abstract: Traffic classification using machine learning continues to be an active research area. The majority of work in this area uses off-the-shelf machine learning tools and treats them as black-box classifiers. This approach turns all the modelling complexity into a feature selection problem. In this paper, we build a problem-specific solution to the traffic classification problem by designing a custom probabilistic graphical model. Graphical models are a modular framework to design classifiers which incorporate domain-specific knowledge. More specifically, our solution introduces semi-supervised learning which means we learn from both labelled and unlabelled traffic flows. We show that our solution performs competitively compared to previous approaches while using less data and simpler features.

Comment: This paper was awarded Honourable Mention for Best Paper at ICML 2009.

Arik Azran. The Rendezvous algorithm: multiclass semi-supervised learning with Markov random walks. In Zoubin Ghahramani, editor, 24th International Conference on Machine Learning, pages 49-56, Corvallis, OR, USA, June 2007. Omnipress.

Abstract: We consider the problem of multiclass classification where both labeled and unlabeled data points are given. We introduce and demonstrate a new approach for estimating a distribution over the missing labels where data points are viewed as nodes of a graph, and pairwise similarities are used to derive a transition probability matrix P for a Markov random walk between them. The algorithm associates each point with a particle which moves between points according to P. Labeled points are set to be absorbing states of the Markov random walk, and the probability of each particle to be absorbed by the different labeled points, as the number of steps increases, is then used to derive a distribution over the associated missing label. A computationally efficient algorithm to implement this is derived and demonstrated on both real and artificial data sets, including a numerical comparison with other methods.

Matthias O. Franz, Younghee Kwon, Carl Edward Rasmussen, and Bernhard Schölkopf. Semi-supervised kernel regression using whitened function classes. In C. E. Rasmussen, H. H. Bülthoff, M. A. Giese, and B. Schölkopf, editors, Lecture Notes in Computer Science (LNCS), volume 3175, pages 18-26, Berlin, Germany, 2004. Springer.

Abstract: The use of non-orthonormal basis functions in ridge regression leads to an often undesired non-isotropic prior in function space. In this study, we investigate an alternative regularization technique that results in an implicit whitening of the basis functions by penalizing directions in function space with a large prior variance. The regularization term is computed from unlabelled input data that characterizes the input distribution. Tests on two datasets using polynomial basis functions showed an improved average performance compared to standard ridge regression.

Non-parametric Bayesian Learning

Non-parametric models are very flexible statistical models in which the complexity of the model grows with the amount of observed data. While traditional parametric models make strong assumptions about how the data was generated, non-parametric models try to make weaker assumptions and let the data "speak for itself". Many non-parametric models can be seen as infinite limits of finite parametric models, and an important family of non-parametric models are derived from Dirichlet processes. See also Gaussian Processes.

Novi Quadrianto, Viktoriia Sharmanska, David A. Knowles, and Zoubin Ghahramani. The supervised ibp: Neighbourhood preserving infinite latent feature models. In 29th Conference on Uncertainty in Artificial Intelligence, Bellevue, USA, July 2013.

Abstract: We propose a probabilistic model to infer supervised latent variables in the Hamming space from observed data. Our model allows simultaneous inference of the number of binary latent variables, and their values. The latent variables preserve neighbourhood structure of the data in a sense that objects in the same semantic concept have similar latent values, and objects in different concepts have dissimilar latent values. We formulate the supervised infinite latent variable problem based on an intuitive principle of pulling objects together if they are of the same type, and pushing them apart if they are not. We then combine this principle with a flexible Indian Buffet Process prior on the latent variables. We show that the inferred supervised latent variables can be directly used to perform a nearest neighbour search for the purpose of retrieval. We introduce a new application of dynamically extending hash codes, and show how to effectively couple the structure of the hash codes with continuously growing structure of the neighbourhood preserving infinite latent feature space.

Comment: arXiv:1302.4245

Konstantina Palla, David A. Knowles, and Zoubin Ghahramani. A nonparametric variable clustering model. In Advances in Neural Information Processing Systems 26, pages 1-9, Lake Tahoe, California, USA, December 2012.

Abstract: Factor analysis models effectively summarise the covariance structure of high dimensional data, but the solutions are typically hard to interpret. This motivates attempting to find a disjoint partition, i.e. a simple clustering, of observed variables into highly correlated subsets. We introduce a Bayesian non-parametric approach to this problem, and demonstrate advantages over heuristic methods proposed to date. Our Dirichlet process variable clustering (DPVC) model can discover block-diagonal covariance structures in data. We evaluate our method on both synthetic and gene expression analysis problems.

Konstantina Palla, David A. Knowles, and Zoubin Ghahramani. An infinite latent attribute model for network data. In 29th International Conference on Machine Learning, Edinburgh, Scotland, June 2012.

Abstract: Latent variable models for network data extract a summary of the relational structure underlying an observed network. The simplest possible models subdivide nodes of the network into clusters; the probability of a link between any two nodes then depends only on their cluster assignment. Currently available models can be classified by whether clusters are disjoint or are allowed to overlap. These models can explain a "flat" clustering structure. Hierarchical Bayesian models provide a natural approach to capture more complex dependencies. We propose a model in which objects are characterised by a latent feature vector. Each feature is itself partitioned into disjoint groups (subclusters), corresponding to a second layer of hierarchy. In experimental comparisons, the model achieves significantly improved predictive performance on social and biological link prediction tasks. The results indicate that models with a single layer hierarchy over-simplify real networks.

Zoubin Ghahramani. Bayesian nonparametrics and the probabilistic approach to modelling. Philosophical Transactions of the Royal Society A, 2012.

Abstract: Modelling is fundamental to many fields of science and engineering. A model can be thought of as a representation of possible data one could predict from a system. The probabilistic approach to modelling uses probability theory to express all aspects of uncertainty in the model. The probabilistic approach is synonymous with Bayesian modelling, which simply uses the rules of probability theory in order to make predictions, compare alternative models, and learn model parameters and structure from data. This simple and elegant framework is most powerful when coupled with flexible probabilistic models. Flexibility is achieved through the use of Bayesian nonparametrics. This article provides an overview of probabilistic modelling and an accessible survey of some of the main tools in Bayesian nonparametrics. The survey covers the use of Bayesian nonparametrics for modelling unknown functions, density estimation, clustering, time series modelling, and representing sparsity, hierarchies, and covariance structure. More specifically it gives brief non-technical overviews of Gaussian processes, Dirichlet processes, infinite hidden Markov models, Indian buffet processes, Kingman's coalescent, Dirichlet diffusion tress, and Wishart processes.

P. Kirk, J. E. Griffin, R. S. Savage, Z. Ghahramani, and D. L. Wild. Bayesian correlated clustering to integrate multiple datasets. Bioinformatics, 2012.

Comment: This paper is available from the Bioinformatics site and a Matlab implementation of MDI is available fromthis site.

Jacob Steinhardt and Zoubin Ghahramani. Flexible martingale priors for deep hierarchies. In 15th International Conference on Artificial Intelligence and Statistics, 2012.

Abstract: When building priors over trees for Bayesian hierarchical models, there is a tension between maintaining desirable theoretical properties such as infinite exchangeability and important practical properties such as the ability to increase the depth of the tree to accommodate new data. We resolve this tension by presenting a family of infinitely exchangeable priors over discrete tree structures that allows the depth of the tree to grow with the data, and then showing that our family contains all hierarchical models with certain mild symmetry properties. We also show that deep hierarchical models are in general intimately tied to a process called a martingale, and use Doob’s martingale convergence theorem to demonstrate some unexpected properties of deep hierarchies.

Kyung-Ah Sohn, Zoubin Ghahramani, and Eric P. Xing. Robust estimation of local genetic ancestry in admixed populations using a non-parametric Bayesian approach. Genetics, 191(4), 2012.

Abstract: We present a new haplotype-based approach for inferring local genetic ancestry of individuals in an admixed population. Most existing approaches for local ancestry estimation ignore the latent genetic relatedness between ancestral populations and treat them as independent. In this paper, we exploit such information by building an inheritance model that describes both the ancestral populations and the admixed population jointly in a unified framework. Based on an assumption that the common hypothetical founder haplotypes give rise to both the ancestral and admixed population haplotypes, we employ an infinite hidden Markov model to characterize each ancestral population and further extend it to generate the admixed population. Through an effective utilization of the population structural information under a principled nonparametric Bayesian framework, the resulting model is significantly less sensitive to the choice and the amount of training data for ancestral populations than state-of-the-arts algorithms. We also improve the robustness under deviation from common modeling assumptions by incorporating population-specific scale parameters that allow variable recombination rates in different populations. Our method is applicable to an admixed population from an arbitrary number of ancestral populations and also performs competitively in terms of spurious ancestry proportions under general multi-way admixture assumption. We validate the proposed method by simulation under various admixing scenarios and present empirical analysis results on worldwide distributed dataset from Human Genome Diversity Project.

Comment: doi: 10.1534/genetics.112.140228

Abstract: We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian processes. This model accommodates input dependent signal and noise correlations between multiple response variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both efficient Markov chain Monte Carlo and variational Bayes inference procedures for this model. We apply GPRN as a multiple output regression and multivariate volatility model, demonstrating substantially improved performance over eight popular multiple output (multi-task) Gaussian process models and three multivariate volatility models on benchmark datasets, including a 1000 dimensional gene expression dataset.

Comment: arXiv:1110.4411

Thomas L. Griffiths and Zoubin Ghahramani. The Indian buffet process: An introduction and review. Journal of Machine Learning Research, 12:1185-1224, April 2011.

Abstract: The Indian buffet process is a stochastic process defining a probability distribution over equivalence classes of sparse binary matrices with a finite number of rows and an unbounded number of columns. This distribution is suitable for use as a prior in probabilistic models that represent objects using a potentially infinite array of features, or that involve bipartite graphs in which the size of at least one class of nodes is unknown. We give a detailed derivation of this distribution, and illustrate its use as a prior in an infinite latent feature model. We then review recent applications of the Indian buffet process in machine learning, discuss its extensions, and summarize its connections to other stochastic processes.

Daniel Hernández-Lobato, José Miguel Hernández-Lobato, and Pierre Dupont. Robust multi-class Gaussian process classification. In Advances in Neural Information Processing Systems 25, 2011.

David A. Knowles and Zoubin Ghahramani. Nonparametric Bayesian sparse factor models with application to gene expression modelling.. Annals of Applied Statistics, 5(2B):1534-1552, 2011.

Abstract: A nonparametric Bayesian extension of Factor Analysis (FA) is proposed where observed data Y is modeled as a linear superposition, G, of a potentially infinite number of hidden factors, X. The Indian Buffet Process (IBP) is used as a prior on G to incorporate sparsity and to allow the number of latent features to be inferred. The model's utility for modeling gene expression data is investigated using randomly generated data sets based on a known sparse connectivity matrix for E. Coli, and on three biological data sets of increasing complexity.

David A. Knowles and Zoubin Ghahramani. Pitman-Yor diffusion trees. In 27nd Conference on Uncertainty in Artificial Intelligence, 2011.

Comment: web site

David A. Knowles, Jurgen Van Gael, and Zoubin Ghahramani. Message passing algorithms for the Dirichlet diffusion tree. In 28th International Conference on Machine Learning, 2011.

Abstract: We demonstrate efficient approximate inference for the Dirichlet Diffusion Tree (Neal, 2003), a Bayesian nonparametric prior over tree structures. Although DDTs provide a powerful and elegant approach for modeling hierarchies they haven't seen much use to date. One problem is the computational cost of MCMC inference. We provide the first deterministic approximate inference methods for DDT models and show excellent performance compared to the MCMC alternative. We present message passing algorithms to approximate the Bayesian model evidence for a specific tree. This is used to drive sequential tree building and greedy search to find optimal tree structures, corresponding to hierarchical clusterings of the data. We demonstrate appropriate observation models for continuous and binary data. The empirical performance of our method is very close to the computationally expensive MCMC alternative on a density estimation problem, and significantly outperforms kernel density estimators.

Comment: web site

David A. Knowles and Thomas P. Minka. Non-conjugate variational message passing for multinomial and binary regression. In Advances in Neural Information Processing Systems 25, 2011.

Comment: web site supplementary

Daniel M. Roy. On the computability and complexity of Bayesian reasoning. In NIPS Workshop on Philosophy and Machine Learning, 2011.

Andrew Gordon Wilson and Zoubin Ghahramani. Generalised Wishart processes. In 27nd Conference on Uncertainty in Artificial Intelligence, 2011.

Comment: Supplementary Material, Best Student Paper Award

Dilan Görür and Carl Edward Rasmussen. Dirichlet process Gaussian mixture models: Choice of the base distribution. Journal of Computer Science and Technology, 25(4):615-625, July 2010, doi 10.1007/s11390-010-9355-8.

Abstract: In the Bayesian mixture modeling framework it is possible to infer the necessary number of components to model the data and therefore it is unnecessary to explicitly restrict the number of components. Nonparametric mixture models sidestep the problem of finding the "correct" number of mixture components by assuming infinitely many components. In this paper Dirichlet process mixture (DPM) models are cast as infinite mixture models and inference using Markov chain Monte Carlo is described. The specification of the priors on the model parameters is often guided by mathematical and practical convenience. The primary goal of this paper is to compare the choice of conjugate and non-conjugate base distributions on a particular class of DPM models which is widely used in applications, the Dirichlet process Gaussian mixture model (DPGMM). We compare computational efficiency and modeling performance of DPGMM defined using a conjugate and a conditionally conjugate base distribution. We show that better density models can result from using a wider class of priors with no or only a modest increase in computational effort.

Sinead Williamson, Katherine A. Heller, C. Wang, and D. M. Blei. The IBP compound Dirichlet process and its application to focused topic modeling. In 27th International Conference on Machine Learning, pages 1151-1158, Haifa, Israel, June 2010.

Abstract: The hierarchical Dirichlet process (HDP) is a Bayesian nonparametric mixed membership model - each data point is modeled with a collection of components of different proportions. Though powerful, the HDP makes an assumption that the probability of a component being exhibited by a data point is positively correlated with its proportion within that data point. This might be an undesirable assumption. For example, in topic modeling, a topic (component) might be rare throughout the corpus but dominant within those documents (data points) where it occurs. We develop the IBP compound Dirichlet process (ICD), a Bayesian nonparametric prior that decouples across-data prevalence and within-data proportion in a mixed membership model. The ICD combines properties from the HDP and the Indian buffet process (IBP), a Bayesian nonparametric prior on binary matrices. The ICD assigns a subset of the shared mixture components to each data point. This subset, the data point's "focus", is determined independently from the amount that each of its components contribute. We develop an ICD mixture model for text, the focused topic model (FTM), and show superior performance over the HDP-based topic model.

Comment: Winner of the Best Paper Award

Sinead Williamson, Peter Orbanz, and Zoubin Ghahramani. Dependent Indian buffet processes. In 13th International Conference on Artificial Intelligence and Statistics, volume 9 of W & CP, pages 924-931, Chia Laguna, Sardinia, Italy, May 2010.

Abstract: Latent variable models represent hidden structure in observational data. To account for the distribution of the observational data changing over time, space or some other covariate, we need generalizations of latent variable models that explicitly capture this dependency on the covariate. A variety of such generalizations has been proposed for latent variable models based on the Dirichlet process. We address dependency on covariates in binary latent feature models, by introducing a dependent Indian Buffet Process. The model generates a binary random matrix with an unbounded number of columns for each value of the covariate. Evolution of the binary matrices over the covariate set is controlled by a hierarchical Gaussian process model. The choice of covariance functions controls the dependence structure and exchangeability properties of the model. We derive a Markov Chain Monte Carlo sampling algorithm for Bayesian inference, and provide experiments on both synthetic and real-world data. The experimental results show that explicit modeling of dependencies significantly improves accuracy of predictions.

R. P. Adams, Zoubin Ghahramani, and Michael I. Jordan. Tree-structured stick breaking for hierarchical data. In Advances in Neural Information Processing Systems 23. The MIT Press, 2010.

Andrew Gordon Wilson and Zoubin Ghahramani. Copula processes. In Advances in Neural Information Processing Systems 23, 2010. Spotlight.

Comment: Supplementary Material, slides.

Finale Doshi-Velez, David Knowles, Shakir Mohamed, and Zoubin Ghahramani. Large scale non-parametric inference: Data parallelisation in the Indian buffet process. In Advances in Neural Information Processing Systems 23, pages 1294-1302, Cambridge, MA, USA, December 2009. The MIT Press.

Abstract: Nonparametric Bayesian models provide a framework for flexible probabilistic modelling of complex datasets. Unfortunately, the high-dimensional averages required for Bayesian methods can be slow, especially with the unbounded representations used by nonparametric models. We address the challenge of scaling Bayesian inference to the increasingly large datasets found in real-world applications. We focus on parallelisation of inference in the Indian Buffet Process (IBP), which allows data points to have an unbounded number of sparse latent features. Our novel MCMC sampler divides a large data set between multiple processors and uses message passing to compute the global likelihoods and posteriors. This algorithm, the first parallel inference scheme for IBP-based models, scales to datasets orders of magnitude larger than have previously been possible.

Finale Doshi-Velez. The Indian buffet process: Scalable inference and extensions. Master's thesis, University of Cambridge, Cambridge, UK, August 2009.

Abstract: Many unsupervised learning problems seek to identify hidden features from observations. In many real-world situations, the number of hidden features is unknown. To avoid specifying the number of hidden features a priori, one can use the Indian Buffet Process (IBP): a nonparametric latent feature model that does not bound the number of active features in a dataset. While elegant, the lack of efficient inference procedures for the IBP has prevented its application in large-scale problems. The core contribution of this thesis are three new inference procedures that allow inference in the IBP to be scaled from a few hundred to 100,000 observations. This thesis contains three parts: (1) An introduction to the IBP and a review of inference techniques and extensions. The first chapters summarise three constructions for the IBP and review all currently published inference techniques. Appendix C reviews extensions of the IBP to date. (2) Novel techniques for scalable Bayesian inference. This thesis presents three new inference procedures: (a) an accelerated Gibbs sampler for efficient Bayesian inference in a broad class of conjugate models, (b) a parallel, asynchronous Gibbs sampler that allows the accelerated Gibbs sampler to be distributed across multiple processors, and (c) a variational inference procedure for the IBP. (3) A framework for structured nonparametric latent feature models. We also present extensions to the IBP to model more sophisticated relationships between the co-occurring hidden features, providing a general framework for correlated non-parametric feature models.

J. Van Gael, A. Vlachos, and Z. Ghahramani. The infinite HMM for unsupervised PoS tagging. In Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 678-687, Singapore, August 2009. Association for Computational Linguistics.

Abstract: We extend previous work on fully unsupervised part-of-speech tagging. Using a non-parametric version of the HMM, called the infinite HMM (iHMM), we address the problem of choosing the number of hidden states in unsupervised Markov models for PoS tagging. We experiment with two non-parametric priors, the Dirichlet and Pitman-Yor processes, on the Wall Street Journal dataset using a parallelized implementation of an iHMM inference algorithm. We evaluate the results with a variety of clustering evaluation metrics and achieve equivalent or better performances than previously reported. Building on this promising result we evaluate the output of the unsupervised PoS tagger as a direct replacement for the output of a fully supervised PoS tagger for the task of shallow parsing and compare the two evaluations.

Comment: This paper was awarded Honourable Mention for Best Paper at ICML 2009.

F. Doshi-Velez and Z. Ghahramani. Correlated non-parametric latent feature models. In Conference on Uncertainty in Artificial Intelligence (UAI 2009), pages 143-150, Montréal, QC, Canada, June 2009. AUAI Press.

Abstract: We are often interested in explaining data through a set of hidden factors or features. To allow for an unknown number of such hidden features, one can use the IBP: a non-parametric latent feature model that does not bound the number of active features in a dataset. However, the IBP assumes that all latent features are uncorrelated, making it inadequate for many real-world problems. We introduce a framework for correlated non-parametric feature models, generalising the IBP. We use this framework to generate several specific models and demonstrate applications on real-world datasets.

F. Doshi-Velez, K.T. Miller, J. Van Gael, and Y.W. Teh. Variational inference for the Indian buffet process. In 12th International Conference on Artificial Intelligence and Statistics, volume 12, pages 137-144, Clearwater Beach, FL, USA, April 2009. Journal of Machine Learning Research.

Abstract: The Indian Buffet Process (IBP) is a nonparametric prior for latent feature models in which observations are influenced by a combination of hidden features. For example, images may be composed of several objects and sounds may consist of several notes. Latent feature models seek to infer these unobserved features from a set of observations; the IBP provides a principled prior in situations where the number of hidden features is unknown. Current inference methods for the IBP have all relied on sampling. While these methods are guaranteed to be accurate in the limit, samplers for the IBP tend to mix slowly in practice. We develop a deterministic variational method for inference in the IBP based on a truncated stick-breaking approximation, provide theoretical bounds on the truncation error, and evaluate our method in several data regimes.

Finale Doshi-Velez, Kurt T. Miller, Jurgen Van Gael, and Yee Whye Teh. Variational inference for the Indian buffet process. Technical Report CBL-2009-001, University of Cambridge, Computational and Biological Learning Laboratory, Department of Engineering, April 2009.

Abstract: The Indian Buffet Process (IBP) is a nonparametric prior for latent feature models in which observations are influenced by a combination of hidden features. For example, images may be composed of several objects and sounds may consist of several notes. Latent feature models seek to infer these unobserved features from a set of observations; the IBP provides a principled prior in situations where the number of hidden features is unknown. Current inference methods for the IBP have all relied on sampling. While these methods are guaranteed to be accurate in the limit, samplers for the IBP tend to mix slowly in practice. We develop a deterministic variational method for inference in the IBP based on truncating to infinite models, provide theoretical bounds on the truncation error, and evaluate our method in several data regimes. This technical report is a longer version of Doshi-Velez et al. (2009).

T. Stepleton, Z. Ghahramani, G. Gordon, and T.-S. Lee. The block diagonal infinite hidden Markov model. In D. van Dyk and M. Welling, editors, 12th International Conference on Artificial Intelligence and Statistics, volume 5, pages 552-559, Clearwater Beach, FL, USA, April 2009. Microtome Publishing (paper) Journal of Machine Learning Research. ISSN 1938-7228.

Abstract: The Infinite Hidden Markov Model (IHMM) extends hidden Markov models to have a countably infinite number of hidden states (Beal et al., 2002; Teh et al., 2006). We present a generalization of this framework that introduces nearly block-diagonal structure in the transitions between the hidden states, where blocks correspond to "sub-behaviors" exhibited by data sequences. In identifying such structure, the model classifies, or partitions, sequence data according to these sub-behaviors in an unsupervised way. We present an application of this model to artificial data, a video gesture classification task, and a musical theme labeling task, and show that components of the model can also be applied to graph segmentation.

Yang Xu, Katherine A. Heller, and Zoubin Ghahramani. Tree-based inference for Dirichlet process mixtures. In D. van Dyk and M. Welling, editors, 12th International Conference on Artificial Intelligence and Statistics, volume 5, pages 623-630, Clearwater Beach, FL, USA, April 2009. Microtome Publishing (paper), Journal of Machine Learning Research (online). ISSN 1938-7228.

Abstract: The Dirichlet process mixture (DPM) is a widely used model for clustering and for general nonparametric Bayesian density estimation. Unfortunately, like in many statistical models, exact inference in a DPM is intractable, and approximate methods are needed to perform efficient inference. While most attention in the literature has been placed on Markov chain Monte Carlo (MCMC) [1, 2, 3], variational Bayesian (VB) [4] and collapsed variational methods [5], [6] recently introduced a novel class of approximation for DPMs based on Bayesian hierarchical clustering (BHC). These tree-based combinatorial approximations efficiently sum over exponentially many ways of partitioning the data and offer a novel lower bound on the marginal likelihood of the DPM [6]. In this paper we make the following contributions: (1) We show empirically that the BHC lower bounds are substantially tighter than the bounds given by VB [4] and by collapsed variational methods [5] on synthetic and real datasets. (2) We also show that BHC offers a more accurate predictive performance on these datasets. (3) We further improve the tree-based lower bounds with an algorithm that efficiently sums contributions from alternative trees. (4) We present a fast approximate method for BHC. Our results suggest that our combinatorial approximate inference methods and lower bounds may be useful not only in DPMs but in other models as well.

Peter Orbanz. Construction of nonparametric Bayesian models from parametric Bayes equations. In Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, editors, Advances in Neural Information Processing Systems 22, pages 1392-1400. The MIT Press, 2009.

Abstract: We consider the general problem of constructing nonparametric Bayesian models on infinite-dimensional random objects, such as functions, infinite graphs or infinite permutations. The problem has generated much interest in machine learning, where it is treated heuristically, but has not been studied in full generality in nonparametric Bayesian statistics, which tends to focus on models over probability distributions. Our approach applies a standard tool of stochastic process theory, the construction of stochastic processes from their finite-dimensional marginal distributions. The main contribution of the paper is a generalization of the classic Kolmogorov extension theorem to conditional probabilities. This extension allows a rigorous construction of nonparametric Bayesian models from systems of finitedimensional, parametric Bayes equations. Using this approach, we show (i) how existence of a conjugate posterior for the nonparametric model can be guaranteed by choosing conjugate finite-dimensional models in the construction, (ii) how the mapping to the posterior parameters of the nonparametric model can be explicitly determined, and (iii) that the construction of conjugate models in essence requires the finite-dimensional models to be in the exponential family. As an application of our constructive framework, we derive a model on infinite permutations, the nonparametric Bayesian analogue of a model recently proposed for the analysis of rank data.

Comment: Supplements (proofs) and techreport version

J. Van Gael, Y.W. Teh, and Z. Ghahramani. The infinite factorial hidden Markov model. In D. Koller, D. Schuurmans, L. Bottou, and Y. Bengio, editors, Advances in Neural Information Processing Systems 21, volume 21, pages 1697-1704, Cambridge, MA, USA, December 2008. The MIT Press.

Abstract: The infinite factorial hidden Markov model is a non-parametric extension of the factorial hidden Markov model. Our model defines a probability distribution over an infinite number of independent binary hidden Markov chains which together produce an observable sequence of random variables. Central to our model is a new type of non-parametric prior distribution inspired by the Indian Buffet Process which we call the Indian Buffet Markov Process.

David Knowles and Zoubin Ghahramani. Infinite sparse factor analysis and infinite independent components analysis. In 7th International Conference on Independent Component Analysis and Signal Separation, pages 381-388, London, UK, September 2007. Springer, doi 10.1007/978-3-540-74494-8_48.

Abstract: A nonparametric Bayesian extension of Independent Components Analysis (ICA) is proposed where observed data Y is modelled as a linear superposition, G, of a potentially infinite number of hidden sources, X. Whether a given source is active for a specific data point is specified by an infinite binary matrix, Z. The resulting sparse representation allows increased data reduction compared to standard ICA. We define a prior on Z using the Indian Buffet Process (IBP). We describe four variants of the model, with Gaussian or Laplacian priors on X and the one or two-parameter IBPs. We demonstrate Bayesian inference under these models using a Markov chain Monte Carlo (MCMC) algorithm on synthetic and gene expression data and compare to standard ICA algorithms.

E. Meeds, Z. Ghahramani, R. Neal, and S.T. Roweis. Modelling dyadic data with binary latent factors. In B. Schölkopf, J. Platt, and T. Hofmann, editors, Advances in Neural Information Processing Systems 19, Bradford Books, pages 977-984, Cambridge, MA, USA, September 2007. The MIT Press. Online contents gives pages 1002-1009, and 977-984 on pdf contents.

Abstract: We introduce binary matrix factorization, a novel model for unsupervised matrix decomposition. The decomposition is learned by fitting a non-parametric Bayesian probabilistic model with binary latent variables to a matrix of dyadic data. Unlike bi-clustering models, which assign each row or column to a single cluster based on a categorical hidden feature, our binary feature model reflects the prior belief that items and attributes can be associated with more than one latent cluster at a time. We provide simple learning and inference rules for this new model and show how to extend it to an infinite model in which the number of features is not a priori fixed but is allowed to grow with the size of the data.

Z. Ghahramani, T.L. Griffiths, and P. Sollich. Bayesian nonparametric latent feature models (with discussion). In J.M. Bernardo, M.J. Bayarri, J.O. Berger, A.P. Dawid, D. Heckerman, A.F.M. Smith, and M. West, editors, Bayesian Statistics 8, pages 201-226, Oxford, UK, July 2007. Oxford University Press.

Abstract: We describe a flexible nonparametric approach to latent variable modelling in which the number of latent variables is unbounded. This approach is based on a probability distribution over equivalence classes of binary matrices with a finite number of rows, corresponding to the data points, and an unbounded number of columns, corresponding to the latent variables. Each data point can be associated with a subset of the possible latent variables, which we refer to as the latent features of that data point. The binary variables in the matrix indicate which latent feature is possessed by which data point, and there is a potentially infinite array of features. We derive the distribution over unbounded binary matrices by taking the limit of a distribution over N×K binary matrices as K→∞. We define a simple generative processes for this distribution which we call the Indian buffet process (IBP; Griffiths and Ghahramani, 2005, 2006) by analogy to the Chinese restaurant process (Aldous, 1985; Pitman, 2002). The IBP has a single hyperparameter which controls both the number of feature per ob ject and the total number of features. We describe a two-parameter generalization of the IBP which has additional flexibility, independently controlling the number of features per object and the total number of features in the matrix. The use of this distribution as a prior in an infinite latent feature model is illustrated, and Markov chain Monte Carlo algorithms for inference are described.

Comment: Includes discussion by David Dunson, and rejoinder.

T. L. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian Buffet Process. In Y. Weiss, B. Schölkopf, and J. Platt, editors, Advances in Neural Information Processing Systems 18, pages 475-482, Cambridge, MA, USA, December 2006. The MIT Press.

Abstract: We define a probability distribution over equivalence classes of binary matrices with a finite number of rows and an unbounded number of columns. This distribution is suitable for use as a prior in probabilistic models that represent objects using a potentially infinite array of features. We identify a simple generative process that results in the same distribution over equivalence classes, which we call the Indian buffet process. We illustrate the use of this distribution as a prior in an infinite latent feature model, deriving a Markov chain Monte Carlo algorithm for inference in this model and applying the algorithm to an image dataset.

Dilan Görür, Frank Jäkel, and Carl Edward Rasmussen. A choice model with infinitely many latent features. In W. W. Cohen and Andrew Moore, editors, 23rd International Conference on Machine Learning, pages 361-368, New York, NY, USA, June 2006. ACM Press, doi 10.1145/1143844.1143890.

Abstract: Elimination by aspects (EBA) is a probabilistic choice model describing how humans decide between several options. The options from which the choice is made are characterized by binary features and associated weights. For instance, when choosing which mobile phone to buy the features to consider may be: long lasting battery, color screen, etc. Existing methods for inferring the parameters of the model assume pre-specified features. However, the features that lead to the observed choices are not always known. Here, we present a non-parametric Bayesian model to infer the features of the options and the corresponding weights from choice data. We use the Indian buffet process (IBP) as a prior over the features. Inference using Markov chain Monte Carlo (MCMC) in conjugate IBP models has been previously described. The main contribution of this paper is an MCMC algorithm for the EBA model that can also be used in inference for other non-conjugate IBP models-this may broaden the use of IBP priors considerably.

Matthew J. Beal, Zoubin Ghahramani, and Carl Edward Rasmussen. The infinite hidden Markov model. In Advances in Neural Information Processing Systems 14, pages 577-584, Cambridge, MA, USA, December 2002. The MIT Press.

Abstract: We show that it is possible to extend hidden Markov models to have a countably infinite number of hidden states. By using the theory of Dirichlet processes we can implicitly integrate out the infinitely many transition parameters, leaving only three hyperparameters which can be learned from data. These three hyperparameters define a hierarchical Dirichlet process capable of capturing a rich set of transition dynamics. The three hyperparameters control the time scale of the dynamics, the sparsity of the underlying state-transition matrix, and the expected number of distinct hidden states in a finite sequence. In this framework it is also natural to allow the alphabet of emitted symbols to be infinite - consider, for example, symbols being possible words appearing in English text.

Carl Edward Rasmussen. The infinite Gaussian mixture model. In Advances in Neural Information Processing Systems 12, pages 554-560. The MIT Press, 2000.

Abstract: In a Bayesian mixture model it is not necessary a priori to limit the number of components to be finite. In this paper an infinite Gaussian mixture model is presented which neatly sidesteps the difficult problem of finding the "right" number of mixture components. Inference in the model is done using an efficient parameter-free Markov Chain that relies entirely on Gibbs sampling.

Approximate Inference

For all but the simplest statistical models, exact learning and inference are computationally intractable. Approximate inference methods make it possible to learn realistic models from large data sets. Generally, approximate inference methods trade off computation time for accuracy. Some of the major classes of approximate inference methods include Markov chain Monte Carlo methods, variational methods and related algorithms such as Expectation Propagation.

David A. Knowles, Jurgen Van Gael, and Zoubin Ghahramani. Message passing algorithms for the Dirichlet diffusion tree. In 28th International Conference on Machine Learning, 2011.

Comment: web site

David Sontag and Daniel M. Roy. The Complexity of Inference in Latent Dirichlet Allocation. In Advances in Neural Information Processing Systems 24, Cambridge, MA, USA, 2011. The MIT Press.

Abstract: We consider the computational complexity of probabilistic inference in Latent Dirichlet Allocation (LDA). First, we study the problem of finding the maximum a posteriori (MAP) assignment of topics to words, where the document's topic distribution is integrated out. We show that, when the effective number of topics per document is small, exact inference takes polynomial time. In contrast, we show that, when a document has a large number of topics, finding the MAP assignment of topics to words in LDA is NP-hard. Next, we consider the problem of finding the MAP topic distribution for a document, where the topic-word assignments are integrated out. We show that this problem is also NP-hard. Finally, we briefly discuss the problem of sampling from the posterior, showing that this is NP-hard in one restricted setting, but leaving open the general question.

Richard E. Turner and Maneesh Sahani. Two problems with variational expectation maximisation for time-series models. In D. Barber, T. Cemgil, and S. Chiappa, editors, Bayesian Time series models, chapter 5, pages 109-130. Cambridge University Press, 2011.

Abstract: Variational methods are a key component of the approximate inference and learning toolbox. These methods fill an important middle ground, retaining distributional information about uncertainty in latent variables, unlike maximum a posteriori methods (MAP), and yet generally requiring less computational time than Monte Carlo Markov Chain methods. In particular the variational Expectation Maximisation (vEM) and variational Bayes algorithms, both involving variational optimisation of a free-energy, are widely used in time-series modelling. Here, we investigate the success of vEM in simple probabilistic time-series models. First we consider the inference step of vEM, and show that a consequence of the well-known compactness property of variational inference is a failure to propagate uncertainty in time, thus limiting the usefulness of the retained distributional information. In particular, the uncertainty may appear to be smallest precisely when the approximation is poorest. Second, we consider parameter learning and analytically reveal systematic biases in the parameters found by vEM. Surprisingly, simpler variational approximations (such a mean-field) can lead to less bias than more complicated structured approximations.

Richard E. Turner. Statistical Models for Natural Sounds. PhD thesis, Gatsby Computational Neuroscience Unit, UCL, 2010.

Pietro Berkes, Richard E. Turner, and Maneesh Sahani. A structured model of video reproduces primary visual cortical organisation. PLoS Computational Biology, 5(9), 09 2009, doi 10.1371/journal.pcbi.1000495.

Abstract: The visual system must learn to infer the presence of objects and features in the world from the images it encounters, and as such it must, either implicitly or explicitly, model the way these elements interact to create the image. Do the response properties of cells in the mammalian visual system reflect this constraint? To address this question, we constructed a probabilistic model in which the identity and attributes of simple visual elements were represented explicitly and learnt the parameters of this model from unparsed, natural video sequences. After learning, the behaviour and grouping of variables in the probabilistic model corresponded closely to functional and anatomical properties of simple and complex cells in the primary visual cortex (V1). In particular, feature identity variables were activated in a way that resembled the activity of complex cells, while feature attribute variables responded much like simple cells. Furthermore, the grouping of the attributes within the model closely parallelled the reported anatomical grouping of simple cells in cat V1. Thus, this generative model makes explicit an interpretation of complex and simple cells as elements in the segmentation of a visual scene into basic independent features, along with a parametrisation of their moment-by-moment appearances. We speculate that such a segmentation may form the initial stage of a hierarchical system that progressively separates the identity and appearance of more articulated visual elements, culminating in view-invariant object recognition.

Jörg Lücke, Richard E. Turner, Maneesh Sahani, and Marc Henniges. Occlusive components analysis. In Y Bengio, D Schuurmans, J Lafferty, C K I Williams, and A Culotta, editors, nips22, pages 1069-1077. mit, 2009.

Abstract: We study unsupervised learning in a probabilistic generative model for occlusion. The model uses two types of latent variables: one indicates which objects are present in the image, and the other how they are ordered in depth. This depth order then determines how the positions and appearances of the objects present, specified in the model parameters, combine to form the image. We show that the object parameters can be learnt from an unlabelled set of images in which objects occlude one another. Exact maximum-likelihood learning is intractable. However, we show that tractable approximations to Expectation Maximization (EM) can be found if the training images each contain only a small number of objects on average. In numerical experiments it is shown that these approximations recover the correct set of object parameters. Experiments on a novel version of the bars test using colored bars, and experiments on more realistic data, show that the algorithm performs well in extracting the generating causes. Experiments based on the standard bars benchmark test for object learning show that the algorithm performs well in comparison to other recent component extraction approaches. The model and the learning algorithm thus connect research on occlusion with the research field of multiple-causes component extraction methods.

J.M. Sung, Z. Ghahramani, and S.Y. Bang. Second-order latent space variational Bayes for approximate Bayesian inference. IEEE Signal Processing Letters, 15:918-921, December 2008.

Abstract: In this letter, we consider a variational approximate Bayesian inference framework, latent-space variational Bayes (LSVB), in the general context of conjugate-exponential family models with latent variables. In the LSVB approach, we integrate out model parameters in an exact way and then perform the variational inference over only the latent variables. It can be shown that LSVB can achieve better estimates of the model evidence as well as the distribution over the latent variables than the popular variational Bayesian expectation-maximization (VBEM). However, the distribution over the latent variables in LSVB has to be approximated in practice. As an approximate implementation of LSVB, we propose a second-order LSVB (SoLSVB) method. In particular, VBEM can be derived as a special case of a first-order approximation in LSVB. SoLSVB can capture higher order statistics neglected in VBEM and can therefore achieve a better approximation. Examples of Gaussian mixture models are used to illustrate the comparison between our method and VBEM, demonstrating the improvement.

J.M. Sung, Z. Ghahramani, and S.Y. Bang. Latent space variational Bayes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(12):2236-2242, November 2008.

Abstract: Variational Bayesian Expectation-Maximization (VBEM), an approximate inference method for probabilistic models based on factorizing over latent variables and model parameters, has been a standard technique for practical Bayesian inference. In this paper, we introduce a more general approximate inference framework for conjugate-exponential family models, which we call Latent-Space Variational Bayes (LSVB). In this approach, we integrate out the model parameters in an exact way, leaving only the latent variables. It can be shown that the LSVB approach gives better estimates of the model evidence as well as the distribution over the latent variables than the VBEM approach, but, in practice, the distribution over the latent variables has to be approximated. As a practical implementation, we present a First-order LSVB (FoLSVB) algorithm to approximate the distribution over the latent variables. From this approximate distribution, one can also estimate the model evidence and the posterior over the model parameters. The FoLSVB algorithm is directly comparable to the VBEM algorithm and has the same computational complexity. We discuss how LSVB generalizes the recently proposed collapsed variational methods to general conjugate-exponential families. Examples based on mixtures of Gaussians and mixtures of Bernoullis with synthetic and real-world data sets are used to illustrate some advantages of our method over VBEM.

Hannes Nickisch and Carl Edward Rasmussen. Approximations for binary Gaussian process classification. Journal of Machine Learning Research, 9:2035-2078, October 2008.

Comment: code.

Pietro Berkes, Richard E. Turner, and Maneesh Sahani. On sparsity and overcompleteness in image models. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, nips20, volume 20. mit, 2008.

Abstract: Computational models of visual cortex, and in particular those based on sparse coding, have enjoyed much recent attention. Despite this currency, the question of how sparse or how over-complete a sparse representation should be, has gone without principled answer. Here, we use Bayesian model-selection methods to address these questions for a sparse-coding model based on a Student-t prior. Having validated our methods on toy data, we find that natural images are indeed best modelled by extremely sparse distributions; although for the Student-t prior, the associated optimal basis size is only modestly over-complete.

J. P. Cunningham. Derivation of Expectation Propagation for "fast Gaussian process methods for point process intensity estimation". Technical report, Stanford University, 2008.

Richard E. Turner and Maneesh Sahani. Modeling natural sounds with modulation cascade processes. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, nips20, volume 20. mit, 2008.

Richard E. Turner and M Sahani. Probabilistic amplitude demodulation. In 7th International Conference on Independent Component Analysis and Signal Separation, pages 544-551, 2007.

Edward Snelson and Zoubin Ghahramani. Compact approximations to Bayesian predictive distributions. In 22nd International Conference on Machine Learning, Bonn, Germany, August 2005. Omnipress.

Abstract: We provide a general framework for learning precise, compact, and fast representations of the Bayesian predictive distribution for a model. This framework is based on minimizing the KL divergence between the true predictive density and a suitable compact approximation. We consider various methods for doing this, both sampling based approximations, and deterministic approximations such as expectation propagation. These methods are tested on a mixture of Gaussians model for density estimation and on binary linear classification, with both synthetic data sets for visualization and several real data sets. Our results show significant reductions in prediction time and memory footprint.

Malte Kuß and Carl Edward Rasmussen. Assessing approximate inference for binary Gaussian process classification. Journal of Machine Learning Research, 6:1679-1704, 2005.

Joaquin Quiñonero-Candela and Carl Edward Rasmussen. A unifying view of sparse approximate Gaussian process regression. Journal of Machine Learning Research, 6:1939-1959, 2005.

Lars Kai Hansen and Carl Edward Rasmussen. Pruning from adaptive regularization. Neural Computation, 6(6):1222-1231, 1994.

Abstract: Inspired by the recent upsurge of interest in Bayesian methods we consider adaptive regularization. A generalization based scheme for adaptation of regularization parameters is introduced and compared to Bayesian regularization. We show that pruning arises naturally within both adaptive regularization schemes. As model example we have chosen the simplest possible: estimating the mean of a random variable with known variance. Marked similarities are found between the two methods in that they both involve a "noise limit", below which they regularize with infinite weight decay, i.e., they prune. However, pruning is not always beneficial. We show explicitly that both methods in some cases may increase the generalization error. This corresponds to situations where the underlying assumptions of the regularizer are poorly matched to the environment.

Bioinformatics

Recent advances in biology have allowed us to collect vast amounts of genetic, proteomic and biomedical data. While this data offers the potential to help us understand the building blocks of life, and to revolutionise medicine, analysing and understanding it poses immense computational and statistical challenges. Our work in Bionformatics includes modelling protein secondary and tertiary structure, analysis of gene microarray data, protein-protein interactions, and biomarker discovery.

P. Kirk, J. E. Griffin, R. S. Savage, Z. Ghahramani, and D. L. Wild. Bayesian correlated clustering to integrate multiple datasets. Bioinformatics, 2012.

Comment: This paper is available from the Bioinformatics site and a Matlab implementation of MDI is available fromthis site.

Kyung-Ah Sohn, Zoubin Ghahramani, and Eric P. Xing. Robust estimation of local genetic ancestry in admixed populations using a non-parametric Bayesian approach. Genetics, 191(4), 2012.

Comment: doi: 10.1534/genetics.112.140228

David A. Knowles and Zoubin Ghahramani. Nonparametric Bayesian sparse factor models with application to gene expression modelling.. Annals of Applied Statistics, 5(2B):1534-1552, 2011.

David A. Knowles, Leopold Parts, Daniel Glass, and John M. Winn. Inferring a measure of physiological age from multiple ageing related phenotypes. In NIPS Workshop: From Statistical Genetics to Predictive Models in Personalized Medicine, 2011.

Abstract: What is ageing? One definition is simultaneous degradation of multiple organ systems. Can an individual be said to be "old" or "young" for their (chronological) age in a scientifically meaningful way? We investigate these questions using ageing related phenotypes measured on the 12,000 female twins in the Twins UK study. We propose a simple linear model of ageing, which allows a latent adjustment to be made to an individual's chronological age to give her "physiological age", shared across the observed phenotypes. We note problems with the analysis resulting from the linearity assumption and show how to alleviate these issues using a non-linear extension. We find more gene expression probes are significantly associated with our measurement of physiological age than to chronological age.

Comment: web site

Mehregan Movassagh, Mun-Kit Choy, David A. Knowles, Lina Cordeddu, Syed Haider, Thomas Down, Lee Siggens, Ana Vujic, Ilenia Simeoni, Chris Penkett, Martin Goddard, Pietro Lio, Martin Bennett, and Roger Foo. Distinct epigenomic features in human cardiomyopathy. Circulation, American Heart Association, 2011.

Abstract: Background. The epigenome refers to marks on the genome including DNA methylation and histone modifications that regulate the expression of underlying genes. A consistent profile of gene expression changes in end- stage cardiomyopathy led us to hypothesise that distinct global patterns of the epigenome may also exist. Methods and Results. We constructed genome-wide maps of DNA methylation and Histone-3 Lysine-36 tri-methylation (H3K36me3)-enrichment for cardiomyopathic and normal human hearts. 506Mb of sequence per library was generated by high-throughput sequencing, covering 24 million out of the 28 million CG di-nucleotides in the human genome. DNA methylation was significantly different in promoter CpG-islands (CGI), intra-genic CGI, gene bodies and H3K36me3-enriched regions of the genome. Moreover DNA methylation differences were present in promoters of upregulated genes but not down-regulated genes. The profile of H3K36me3-enrichment itself was also significantly different in protein-coding regions of the genome. Conclusions. Distinct epigenomic patterns exist in important DNA elements of the human cardiac genome in end-stage cardiomyopathy. If epigenomic patterns track with disease progression, assays for the epigenome may be more useful than quantification of mRNA for assessing prognosis in heart failure. These results open up an important new horizon of research and further studies will be needed to determine how epigenomics contribute to altered gene expression in cardiomyopathy.

Cornelia Schone, Anne Venner, David A. Knowles, Mahesh M Karnani, and Denis Burdakov. Dichotomous cellular properties of mouse orexin/hypocretin neurons. The Journal of Physiology, 2011.

Abstract: Hypothalamic hypocretin/orexin (hcrt/orx) neurons recently emerged as critical regulators of sleep-wake cycles, reward-seeking, and body energy balance. However, at the level of cellular and network properties, it remains unclear whether hcrt/orx neurons are one homogenous population, or whether there are several distinct types of hcrt/orx cells. Here, we collated diverse structural and functional information about individual hcrt/orx neurons in mouse brain slices, by combining patch-clamp analysis of spike firing, membrane currents, and synaptic inputs with confocal imaging of cell shape and subsequent 3-dimensional Sholl analysis of dendritic architecture. Statistical cluster analysis of intrinsic firing properties revealed that hcrt/orx neurons fall into two distinct types. These two cell types also differ in the complexity of their dendritic arbour, the strength of AMPA and GABAA receptor-mediated synaptic drive that they receive, and the density of low-threshold, 4-aminopyridine-sensitive, transient K+ current. Our results provide quantitative evidence that, at the cellular level, the mouse hcrt/orx system is composed of two classes of neurons with different firing properties, morphologies, and synaptic input organization.

Daniel Glass, Leopold Parts, David A. Knowles, Abraham Aviv, , and Tim D. Spector. No correlation between childhood maltreatment and telomere length.. Biological Psychiatry, 68(6):21-22, 2010.

Abstract: Telomeres are lengths of repetitive DNA that cap the ends of chromosomes. They protect the ends of the chromosome and shorten with each cell division. Short leukocyte telomere length has been related to a number of age-related diseases. In addition, shorter telomere length has been associated with environmental factors such as smoking and lack of exercise. In a recent issue of Biological Psychiatry, Tyrka et al. (4) published a report suggesting a link between maltreatment in childhood and telomere shortening in 31 subjects. Individuals who had suffered maltreatment had telomere length .70 +/- .24 compared with 1.02 +/- .52 in individuals who had not been abused.

David A. Knowles, Leopold Parts, Daniel Glass, and John M. Winn. Modeling skin and ageing phenotypes using latent variable models in infer.net. In NIPS Workshop: Predictive Models in Personalized Medicine Workshop, 2010.

Abstract: We demonstrate and compare three unsupervised Bayesian latent variable models implemented in Infer.NET for biomedical data modeling of 42 skin and ageing phenotypes measured on the 12,000 female twins in the Twins UK study. We address various data modeling problems include high missingness, heterogeneous data, and repeat observations. We compare the proposed models in terms of their performance at predicting disease labels and symptoms from available explanatory variables, concluding that factor analysis type models have the strongest statistical performance in this setting. We show that such models can be combined with regression components for improved interpretability.

Comment: web site

C. Lippert, Z. Ghahramani, and K. Borgwardt. Gene function prediction from synthetic lethality networks via ranking on demand. Bioinformatics, 26:912-918, 2010.

Abstract: Motivation: Synthetic lethal interactions represent pairs of genes whose individual mutations are not lethal, while the double mutation of both genes does incur lethality. Several studies have shown a correlation between functional similarity of genes and their distances in networks based on synthetic lethal interactions. However, there is a lack of algorithms for predicting gene function from synthetic lethality interaction networks.
Results: In this article, we present a novel technique called kernelROD for gene function prediction from synthetic lethal interaction networks based on kernel machines. We apply our novel algorithm to Gene Ontology functional annotation prediction in yeast. Our experiments show that our method leads to improved gene function prediction compared with state-of-the-art competitors and that combining genetic and congruence networks leads to a further improvement in prediction accuracy.

O. Stegle, K. J. Denby, E. J. Cooke, D. L. Wild, Z. Ghahramani, and K. M. Borgwardt. A robust Bayesian two-sample test for detecting intervals of differential gene expression in microarray time series. Journal of Computational Biology, 17(3):1-13, 2010, doi 10.1089/cmb.2009.0175.

Abstract: Understanding the regulatory mechanisms that are responsible for an organism's response to environmental change is an important issue in molecular biology. A first and important step towards this goal is to detect genes whose expression levels are affected by altered external conditions. A range of methods to test for differential gene expression, both in static as well as in time-course experiments, have been proposed. While these tests answer the question whether a gene is differentially expressed, they do not explicitly address the question when a gene is differentially expressed, although this information may provide insights into the course and causal structure of regulatory programs. In this article, we propose a twosample test for identifying intervals of differential gene expression in microarray time series. Our approach is based on Gaussian process regression, can deal with arbitrary numbers of replicates, and is robust with respect to outliers. We apply our algorithm to study the response of Arabidopsis thaliana genes to an infection by a fungal pathogen using a microarray time series dataset covering 30,336 gene probes at 24 observed time points. In classification experiments, our test compares favorably with existing methods and provides additional insights into time-dependent differential expression.

R. S. Savage, Z. Ghahramani, J. E. Griffin, B. de la Cruz, and D. L. Wild. Discovering transcriptional modules by Bayesian data integration. Bioinformatics, 26:i158-i167, 2010.

Abstract: Motivation: We present a method for directly inferring transcriptional modules (TMs) by integrating gene expression and transcription factor binding (ChIP-chip) data. Our model extends a hierarchical Dirichlet process mixture model to allow data fusion on a gene-by-gene basis. This encodes the intuition that co-expression and co-regulation are not necessarily equivalent and hence we do not expect all genes to group similarly in both datasets. In particular, it allows us to identify the subset of genes that share the same structure of transcriptional modules in both datasets.
Results: We find that by working on a gene-by-gene basis, our model is able to extract clusters with greater functional coherence than existing methods. By combining gene expression and transcription factor binding (ChIP-chip) data in this way, we are better able to determine the groups of genes that are most likely to represent underlying TMs.
Availability: If interested in the code for the work presented in this article, please contact the authors.

R. Silva, K. A. Heller, Z. Ghahramani, and E. M. Airoldi. Ranking relations using analogies in biological and information networks. Annals of Applied Statistics, 4(2):615-644, 2010.

Abstract: Analogical reasoning depends fundamentally on the ability to learn and generalize about relations between objects. We develop an approach to relational learning which, given a set of pairs of objects S = A(1):B(1), A(2):B(2), ..., A(N):B(N), measures how well other pairs A:B fit in with the set S. Our work addresses the question: is the relation between objects A and B analogous to those relations found in S? Such questions are particularly relevant in information retrieval, where an investigator might want to search for analogous pairs of objects that match the query set of interest. There are many ways in which objects can be related, making the task of measuring analogies very challenging. Our approach combines a similarity measure on function spaces with Bayesian analysis to produce a ranking. It requires data containing features of the objects of interest and a link matrix specifying which relationships exist; no further attributes of such relationships are necessary. We illustrate the potential of our method on text analysis and information networks. An application on discovering functional interactions between pairs of proteins is discussed in detail, where we show that our approach can work in practice even if a small set of protein pairs is provided.

O. Stegle, K. Denby, S. McHattie, A. Meade, D. Wild, Z. Ghahramani, and K Borgwardt. Discovering temporal patterns of differential gene expression in microarray time series. In German Conference on Bioinformatics, pages 133-142, Halle, Germany, September 2009.

Abstract: A wealth of time series of microarray measurements have become available over recent years. Several two-sample tests for detecting differential gene expression in these time series have been defined, but they can only answer the question whether a gene is differentially expressed across the whole time series, not in which intervals it is differentially expressed. In this article, we propose a Gaussian process based approach for studying these dynamics of differential gene expression. In experiments on Arabidopsis thaliana gene expression levels, our novel technique helps us to uncover that the family of WRKY transcription factors appears to be involved in the early response to infection by a fungal pathogen.

C. Lippert, O. Stegle, Z. Ghahramani, and K. Borgwardt. A kernel method for unsupervised structured network inference. In D. van Dyk and M. Welling, editors, 12th International Conference on Artificial Intelligence and Statistics, volume 5, pages 368-375, Clearwater Beach, FL, USA, April 2009. Journal of Machine Learning Research. ISSN: 1938-7228.

Abstract: Network inference is the problem of inferring edges between a set of real-world objects, for instance, interactions between pairs of proteins in bioinformatics. Current kernel-based approaches to this problem share a set of common features: (i) they are supervised and hence require labeled training data; (ii) edges in the network are treated as mutually independent and hence topological properties are largely ignored; (iii) they lack a statistical interpretation. We argue that these common assumptions are often undesirable for network inference, and propose (i) an unsupervised kernel method (ii) that takes the global structure of the network into account and (iii) is statistically motivated. We show that our approach can explain commonly used heuristics in statistical terms. In experiments on social networks, dfferent variants of our method demonstrate appealing predictive performance.

David A. Knowles and Susan Holmes. Statistical tools for ultra-deep pyrosequencing of fast evolving viruses. In NIPS Workshop: Computational Biology, 2009.

Abstract: We aim to detect minor variant Hepatitis B viruses (HBV) in 38 pyrosequencing samples from infected individuals. Errors involved in the amplification and ultra deep pyrosequencing (UDPS) of these samples are characterised using HBV plasmid controls. Homopolymeric regions and quality scores are found to be significant covariates in determining insertion and deletion (indel) error rates, but not mismatch rates which depend on the nucleotide transition matrix. This knowledge is used to derive two methods for classifying genuine mutations: a hypothesis testing framework and a mixture model. Using an approximate "ground truth" from a limiting dilution Sanger sequencing run, these methods are shown to outperform the naive percentage threshold approach. The possibility of early stage PCR errors becoming significant is investigated by simulation, which underlines the importance of the initial copy number.

Comment: web site

O. Stegle, K. Denby, David L. Wild, Zoubin Ghahramani, and Karsten Borgwardt. A robust Bayesian two-sample test for detecting intervals of differential gene expression in microarray time series. In 13th Annual International Conference on Research in Computational Molecular Biology (RECOMB 2009), volume 5541 of Lecture Notes in Bioinformatics, pages 201-216, Tucson, AZ, USA, 2009. Springer-Verlag, doi 10.1007/978-3-642-02008-7_14.

Abstract: Understanding the regulatory mechanisms that are responsible for an organism's response to environmental changes is an important question in molecular biology. A first and important step towards this goal is to detect genes whose expression levels are affected by altered external conditions. A range of methods to test for differential gene expression, both in static as well as in time-course experiments, have been proposed. While these tests answer the question whether a gene is differentially expressed, they do not explicitly address the question when a gene is differentially expressed, although this information may provide insights into the course and causal structure of regulatory programs. In this article, we propose a two-sample test for identifying intervals of differential gene expression in microarray time series. Our approach is based on Gaussian process regression, can deal with arbitrary numbers of replicates and is robust with respect to outliers. We apply our algorithm to study the response of Arabidopsis thaliana genes to an infection by a fungal pathogen using a microarray time series dataset covering 30,336 gene probes at 24 time points. In classification experiments our test compares favorably with existing methods and provides additional insights into time-dependent differential expression.

Information Retrieval

Information retrieval concerns develping systems that find material from within a large unstructured collection (e.g. the internet) that satisfy the user's need. The best example of such systems are web search engines, such as Google, but there are many other specialized applications of information retrieval (such as collaborative filtering and recommender systems). Information retrieval can be thought of as an inference problem: given the user's query, what are the relevant items in the data collection?

R. Silva, K. A. Heller, Z. Ghahramani, and E. M. Airoldi. Ranking relations using analogies in biological and information networks. Annals of Applied Statistics, 4(2):615-644, 2010.

Reinforcement Learning and Control

We are interested in understanding the human sensory motor system from a mathematical, computational and engineering point of view. To do this, we need to use concepts from control theory, optimization, machine learning and statistics, as well as experimental methods based on human psychophysics and virtual reality. These formal tools are also useful for advancing robotics and decision theory.

Comment: accepted

Marc Peter Deisenroth, Carl Edward Rasmussen, and Dieter Fox. Learning to control a low-cost manipulator using data-efficient reinforcement learning. In 9th International Conference on Robotics: Science & Systems, Los Angeles, CA, USA, June 2011.

Abstract: Over the last years, there has been substantial progress in robust manipulation in unstructured environments. The long-term goal of our work is to get away from precise, but very expensive robotic systems and to develop affordable, potentially imprecise, self-adaptive manipulator systems that can interactively perform tasks such as playing with children. In this paper, we demonstrate how a low-cost off-the-shelf robotic system can learn closed-loop policies for a stacking task in only a handful of trials - from scratch. Our manipulator is inaccurate and provides no pose feedback. For learning a controller in the work space of a Kinect-style depth camera, we use a model-based reinforcement learning technique. Our learning method is data efficient, reduces model bias, and deals with several noise sources in a principled way during long-term planning. We present a way of incorporating state-space constraints into the learning process and analyze the learning gain by exploiting the sequential structure of the stacking task.

Comment: project site

Daniel A. Braun, Pedro A. Ortega, Evangelos Theodorou, and Stefan Schaal. Path integral control and bounded rationality. In 2011 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, 2011.

Abstract: Path integral methods have recently been shown to be applicable to a very general class of optimal control problems. Here we examine the path integral formalism from a decision-theoretic point of view, since an optimal controller can always be regarded as an instance of a perfectly rational decision-maker that chooses its actions so as to maximize its expected utility. The problem with perfect rationality is, however, that finding optimal actions is often very difficult due to prohibitive computational resource costs that are not taken into account. In contrast, a bounded rational decision-maker has only limited resources and therefore needs to strike some compromise between the desired utility and the required resource costs. In particular, we suggest an information-theoretic measure of resource costs that can be derived axiomatically. As a consequence we obtain a variational principle for choice probabilities that trades off maximizing a given utility criterion and avoiding resource costs that arise due to deviating from initially given default choice probabilities. The resulting bounded rational policies are in general probabilistic. We show that the solutions found by the path integral formalism are such bounded rational policies. Furthermore, we show that the same formalism generalizes to discrete control problems, leading to linearly solvable bounded rational control policies in the case of Markov systems. Importantly, Bellman's optimality principle is not presupposed by this variational principle, but it can be derived as a limit case. This suggests that the information theoretic formalization of bounded rationality might serve as a general principle in control design that unifies a number of recently reported approximate optimal control methods both in the continuous and discrete domain.

Marc Peter Deisenroth and Carl Edward Rasmussen. PILCO: A model-based and data-efficient approach to policy search. In 28th International Conference on Machine Learning, 2011.

Comment: web site

Finale Doshi-Velez and Zoubin Ghahramani. A comparison of human and agent reinforcement learning in partially observable domains. In 33rd Annual Meeting of the Cognitive Science Society, Boston, MA, 2011.

Abstract: It is commonly stated that reinforcement learning (RL) algorithms learn slower than humans. In this work, we investigate this claim using two standard problems from the RL literature. We compare the performance of human subjects to RL techniques. We find that context-the meaningfulness of the observations—-plays a significant role in the rate of human RL. Moreover, without contextual information, humans often fare much worse than classic algorithms. Comparing the detailed responses of humans and RL algorithms, we also find that humans appear to employ rather different strategies from standard algorithms, even in cases where they had indistinguishable performance to them. Our research both sheds light on human RL and provides insights for improving RL algorithms.

Joseph Hall, Carl Edward Rasmussen, and Jan Maciejowski. Reinforcement learning with reference tracking control in continuous state spaces. In Proceedings of 50th IEEE Conference on Decision and Control and European Control Conference, 2011.

Abstract: The contribution described in this paper is an algorithm for learning nonlinear, reference tracking, control policies given no prior knowledge of the dynamical system and limited interaction with the system through the learning process. Concepts from the field of reinforcement learning, Bayesian statistics and classical control have been brought together in the formulation of this algorithm which can be viewed as a form indirect self tuning regulator. On the task of reference tracking using the inverted pendulum it was shown to yield generally improved performance on the best controller derived from the standard linear quadratic method using only 30 s of total interaction with the system. Finally, the algorithm was shown to work on the double pendulum proving its ability to solve nontrivial control tasks.

Pedro A. Ortega and Daniel A. Braun. Information, utility and bounded rationality. In The fourth conference on artificial general intelligence, volume 6830 of Lecture Notes on Artificial Intelligence, pages 269-274. Springer-Verlag, 2011.

Abstract: Perfectly rational decision-makers maximize expected utility, but crucially ignore the resource costs incurred when determining optimal actions. Here we employ an axiomatic framework for bounded rational decision-making based on a thermodynamic interpretation of resource costs as information costs. This leads to a variational free utility principle akin to thermodynamical free energy that trades off utility and information costs. We show that bounded optimal control solutions can be derived from this variational principle, which leads in general to stochastic policies. Furthermore, we show that risk-sensitive and robust (minimax) control schemes fall out naturally from this framework if the environment is considered as a bounded rational and perfectly rational opponent, respectively. When resource costs are ignored, the maximum expected utility principle is recovered.

Pedro A. Ortega, Daniel A. Braun, and Simon Godsill. Reinforcement learning and the Bayesian control rule. In The fourth conference on artificial general intelligence, volume 6830 of Lecture Notes on Artificial Intelligence, pages 281-285. Springer-Verlag, 2011.

Abstract: We present an actor-critic scheme for reinforcement learning in complex domains. The main contribution is to show that planning and I/O dynamics can be separated such that an intractable planning problem reduces to a simple multi-armed bandit problem, where each lever stands for a potentially arbitrarily complex policy. Furthermore, we use the Bayesian control rule to construct an adaptive bandit player that is universal with respect to a given class of optimal bandit players, thus indirectly constructing an adaptive agent that is universal with respect to a given class of policies.

Pedro A. Ortega. A Unified Framework for Resource-Bounded Agents Interacting with an Unknown Environment. PhD thesis, Department of Engineering, University of Cambridge, 2011.

Abstract: The aim of this thesis is to present a mathematical framework for conceptualizing and constructing adaptive autonomous systems under resource constraints. The first part of this thesis contains a concise presentation of the foundations of classical agency: namely the formalizations of decision making and learning. Decision making includes: (a) subjective expected utility (SEU) theory, the framework of decision making under uncertainty; (b) the maximum SEU principle to choose the optimal solution; and (c) its application to the design of autonomous systems, culminating in the Bellman optimality equations. Learning includes: (a) Bayesian probability theory, the theory for reasoning under uncertainty that extends logic; and (b) Bayes-Optimal agents, the application of Bayesian probability theory to the design of optimal adaptive agents. Then, two major problems of the maximum SEU principle are highlighted: (a) the prohibitive computational costs and (b) the need for the causal precedence of the choice of the policy. The second part of this thesis tackles the two aforementioned problems. First, an information-theoretic notion of resources in autonomous systems is established. Second, a framework for resource-bounded agency is introduced. This includes: (a) a maximum bounded SEU principle that is derived from a set of axioms of utility; (b) an axiomatic model of probabilistic causality, which is applied for the formalization of autonomous systems having uncertainty over their policy and environment; and (c) the Bayesian control rule, which is derived from the maximum bounded SEU principle and the model of causality, implementing a stochastic adaptive control law that deals with the case where autonomous agents are uncertain about their policy and environment.

Daniel A. Braun and Pedro A. Ortega. A minimum relative entropy principle for adaptive control in linear quadratic regulators. In Proceedings of the 7th international conference on informatics in control, automation and robotics, page (in press), 2010.

Abstract: The design of optimal adaptive controllers is usually based on heuristics, because solving Bellman's equations over information states is notoriously intractable. Approximate adaptive controllers often rely on the principle of certainty-equivalence where the control process deals with parameter point estimates as if they represented ``true'' parameter values. Here we present a stochastic control rule instead where controls are sampled from a posterior distribution over a set of probabilistic input-output models and the true model is identified by Bayesian inference. This allows reformulating the adaptive control problem as an inference and sampling problem derived from a minimum relative entropy principle. Importantly, inference and action sampling both work forward in time and hence such a Bayesian adaptive controller is applicable on-line. We demonstrate the improved performance that can be achieved by such an approach for linear quadratic regulator examples.

Marc Peter Deisenroth. Efficient reinforcement learning using Gaussian processes. PhD thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2010.

Pedro A. Ortega and Daniel A. Braun. An axiomatic formalization of bounded rationality based on a utility-information equivalence. Technical report, Dept. of Engineering, University of Cambridge, 2010.

Abstract: Classic decision-theory is based on the maximum expected utility (MEU) principle, but crucially ignores the resource costs incurred when determining optimal decisions. Here we propose an axiomatic framework for bounded decision-making that considers resource costs. Agents are formalized as probability measures over input-output streams. We postulate that any such probability measure can be assigned a corresponding conjugate utility function based on three axioms: utilities should be real-valued, additive and monotonic mappings of probabilities. We show that these axioms enforce a unique conversion law between utility and probability (and thereby, information). Moreover, we show that this relation can be characterized as a variational principle: given a utility function, its conjugate probability measure maximizes a free utility functional. Transformations of probability measures can then be formalized as a change in free utility due to the addition of new constraints expressed by a target utility function. Accordingly, one obtains a criterion to choose a probability measure that trades off the maximization of a target utility function and the cost of the deviation from a reference distribution. We show that optimal control, adaptive estimation and adaptive control problems can be solved this way in a resource-efficient way. When resource costs are ignored, the MEU principle is recovered. Our formalization might thus provide a principled approach to bounded rationality that establishes a close link to information theory.

Pedro A. Ortega and Daniel A. Braun. A Bayesian rule for adaptive control based on causal interventions. In The third conference on artificial general intelligence, pages 115-120, Paris, 2010. Atlantis Press.

Abstract: Explaining adaptive behavior is a central problem in artificial intelligence research. Here we formalize adaptive agents as mixture distributions over sequences of inputs and outputs (I/O). Each distribution of the mixture constitutes a "possible world", but the agent does not know which of the possible worlds it is actually facing. The problem is to adapt the I/O stream in a way that is compatible with the true world. A natural measure of adaptation can be obtained by the Kullback Leibler (KL) divergence between the I/O distribution of the true world and the I/O distribution expected by the agent that is uncertain about possible worlds. In the case of pure input streams, the Bayesian mixture provides a well-known solution for this problem. We show, however, that in the case of I/O streams this solution breaks down, because outputs are issued by the agent itself and require a different probabilistic syntax as provided by intervention calculus. Based on this calculus, we obtain a Bayesian control rule that allows modeling adaptive behavior with mixture distributions over I/O streams. This rule might allow for a novel approach to adaptive control based on a minimum KL-principle.

Pedro A. Ortega and Daniel A. Braun. A conversion between utility and information. In The third conference on artificial general intelligence, pages 115-120, Paris, 2010. Atlantis Press.

Abstract: Rewards typically express desirabilities or preferences over a set of alternatives. Here we propose that rewards can be defined for any probability distribution based on three desiderata, namely that rewards should be real- valued, additive and order-preserving, where the later implies that more probable events should also be more desirable. Our main result states that rewards are then uniquely determined by the negative information content. To analyze stochastic processes, we define the utility of a realization as its reward rate. Under this interpretation, we show that the expected utility of a stochastic process is its negative entropy rate. Furthermore, we apply our results to analyze agent-environment interactions. We show that the expected utility that will actually be achieved by the agent is given by the negative cross-entropy from the input-output (I/O) distribution of the coupled interaction system and the agent's I/O distribution. Thus, our results allow for an information-theoretic interpretation of the notion of utility and the characterization of agent-environment interactions in terms of entropy dynamics.

Pedro A. Ortega and Daniel A. Braun. A minimum relative entropy principle for learning and acting. Journal of Artificial Intelligence Research, 38:475-511, 2010, doi 10.1613/jair.3062.

Abstract: This paper proposes a method to construct an adaptive agent that is universal with respect to a given class of experts, where each expert is designed specifically for a particular environment. This adaptive control problem is formalized as the problem of minimizing the relative entropy of the adaptive agent from the expert that is most suitable for the unknown environment. If the agent is a passive observer, then the optimal solution is the well-known Bayesian predictor. However, if the agent is active, then its past actions need to be treated as causal interventions on the I/O stream rather than normal probability conditions. Here it is shown that the solution to this new variational problem is given by a stochastic controller called the Bayesian control rule, which implements adaptive behavior as a mixture of experts. Furthermore, it is shown that under mild assumptions, the Bayesian control rule converges to the control law of the most suitable expert.

Marc Peter Deisenroth and Carl Edward Rasmussen. Efficient reinforcement learning for motor control. In 10th International PhD Workshop on Systems and Control, Hluboká nad Vltavou, Czech Republic, September 2009.

Abstract: Artificial learners often require many more trials than humans or animals when learning motor control tasks in the absence of expert knowledge. We implement two key ingredients of biological learning systems, generalization and incorporation of uncertainty into the decision-making process, to speed up artificial learning. We present a coherent and fully Bayesian framework that allows for efficient artificial learning in the absence of expert knowledge. The success of our learning framework is demonstrated on challenging nonlinear control problems in simulation and in hardware.

Marc Peter Deisenroth and Carl Edward Rasmussen. Bayesian inference for efficient learning in control. In Multidisciplinary Symposium on Reinforcement Learning, Montréal, QC, Canada, June 2009.

Abstract: In contrast to humans or animals, artificial learners often require more trials when learning motor control tasks solely based on experience. Efficient autonomous learners will reduce the amount of engineering required to solve control problems. By using probabilistic forward models, we can employ two key ingredients of biological learning systems to speed up artificial learning. We present a consistent and coherent Bayesian framework that allows for efficient autonomous experience-based learning. We demonstrate the success of our learning algorithm by applying it to challenging nonlinear control problems in simulation and in hardware.

Marc Peter Deisenroth, Carl Edward Rasmussen, and Jan Peters. Gaussian process dynamic programming. Neurocomputing, 72(7-9):1508-1524, March 2009, doi 10.1016/j.neucom.2008.12.019.

Comment: code.

Carl Edward Rasmussen and Marc Peter Deisenroth. Probabilistic inference for fast learning in control. In S. Girgin, M. Loth, R. Munos, P. Preux, and D. Ryabko, editors, Recent Advances in Reinforcement Learning, volume 5323 of Lecture Notes in Computer Science (LNCS), pages 229-242, Villeneuve d'Ascq, France, November 2008. Springer-Verlag.

Abstract: We provide a novel framework for very fast model-based reinforcement learning in continuous state and action spaces. The framework requires probabilistic models that explicitly characterize their levels of confidence. Within this framework, we use flexible, non-parametric models to describe the world based on previously collected experience. We demonstrate learning on the cart-pole problem in a setting where we provide very limited prior knowledge about the task. Learning progresses rapidly, and a good policy is found after only a hand-full of iterations.

Comment: videos and more. slides.

Comment: code. slides

Carl Edward Rasmussen and Malte Kuß. Gaussian processes in reinforcement learning. In S. Thrun, L.K. Saul, and B. Schölkopf, editors, Advances in Neural Information Processing Systems 16, pages 751-759, Cambridge, MA, USA, December 2004. The MIT Press.

Abstract: We exploit some useful properties of Gaussian process (GP) regression models for reinforcement learning in continuous state spaces and discrete time. We demonstrate how the GP model allows evaluation of the value function in closed form. The resulting policy iteration algorithm is demonstrated on a simple problem with a two dimensional state space. Further, we speculate that the intrinsic ability of GP models to characterise distributions of functions would allow the method to capture entire distributions over future values instead of merely their expectation, which has traditionally been the focus of much of reinforcement learning.

Juš Kocijan, Roderick Murray-Smith, Carl Edward Rasmussen, and Agathe Girard. Gaussian process model based predictive control. In American Control Conference, pages 2214-2219, 2004.

Time Series Models

Modelling time series and sequential data is an essential part of many different areas of science and engineering, including for example, signal processing and control, bioinformatics, speech recognition, econometrics and finance. Using basic building blocks such as hidden Markov models, linear Gaussian state-space models, and Bayesian networks, it is possible to develop sophisticated time series models for real world data. However learning (parameter inference / system identification) becomes computationally challenging for such sophisticated models.

Creighton Heaukulani and Zoubin Ghahramani. Dynamic probabilistic models for latent feature propagation in social networks. In 30th International Conference on Machine Learning, Atlanta, Georgia, USA, June 2013.

Abstract: Current Bayesian models for dynamic social network data have focused on modelling the influence of evolving unobserved structure on observed social interactions. However, an understanding of how observed social relationships from the past affect future unobserved structure in the network has been neglected. In this paper, we introduce a new probabilistic model for capturing this phenomenon, which we call latent feature propagation, in social networks. We demonstrate our model's capability for inferring such latent structure in varying types of social network datasets, and experimental studies show this structure achieves higher predictive performance on link prediction and forecasting tasks.

Comment: arXiv:1302.4245

Ryan D. Turner and Carl Edward Rasmussen. Model based learning of sigma points in unscented Kalman filtering. Neurocomputing, 80:47-53, 2012, doi 10.1016/j.neucom.2011.07.029.

J. H. Macke, L. Busing, J. P. Cunningham, B. M. Yu, K. V. Shenoy, and M. Sahani. Empirical models of spiking in neural populations. In Advances in Neural Information Processing Systems 25, pages 1-8, Granada, Spain, December 2011.

Abstract: Neurons in the neocortex code and compute as part of a locally interconnected population. Large-scale multi-electrode recording makes it possible to access these population processes empirically by fitting statistical models to unaveraged data. What statistical structure best describes the concurrent spiking of cells within a local network? We argue that in the cortex, where firing exhibits extensive correlations in both time and space and where a typical sample of neurons still reflects only a very small fraction of the local population, the most appropriate model captures shared variability by a low-dimensional latent process evolving with smooth dynamics, rather than by putative direct coupling. We test this claim by comparing a latent dynamical model with realistic spiking observations to coupled generalised linear spike-response models (GLMs) using cortical recordings. We find that the latent dynamical approach outperforms the GLM in terms of goodness-of- fit, and reproduces the temporal correlations in the data more accurately. We also compare models whose observations models are either derived from a Gaussian or point-process models, finding that the non-Gaussian model provides slightly better goodness-of-fit and more realistic population spike counts.

B. Petreska, B. M. Yu, J. P. Cunningham, G. Santhanam, S. I. Ryu, K. V. Shenoy, and M. Sahani. Dynamical segmentation of single trials from population neural data. In Advances in Neural Information Processing Systems 25, pages 1-8, Granada, Spain, December 2011.

Abstract: Simultaneous recordings of many neurons embedded within a recurrently-connected cortical network may provide concurrent views into the dynamical processes of that network, and thus its computational function. In principle, these dynamics might be identified by purely unsupervised, statistical means. Here, we show that a Hidden Switching Linear Dynamical Systems (HSLDS) model - in which multiple linear dynamical laws approximate and nonlinear and potentially non-stationary dynamical process - is able to distinguish dynamical regimes within single-trial motor cortical activity associated with the preparation and initiation of hand movements. The regimes are identified without reference to behavioural or experimental epochs, but nonetheless transitions between them correlate strongly with external events whose timing may vary from trial to trial. The HSLDS model also performs better than recent comparable models in predicting the firing rate of an isolated neuron based on the firing rates of others, suggesting that it captures more of the "Shared variance" of the data. Thus, the method is able to trace the dynamical processes underlying the coordinated evolution of network activity in a way that appears to reflect its computational role.

Abstract: We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian processes. This model accommodates input dependent signal and noise correlations between multiple response variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both efficient Markov chain Monte Carlo and variational Bayes inference procedures for this model. We apply GPRN as a multiple output regression and multivariate volatility model, demonstrating substantially improved performance over eight popular multiple output (multi-task) Gaussian process models and three multivariate volatility models on benchmark datasets, including a 1000 dimensional gene expression dataset.

Comment: arXiv:1110.4411

J. P. Cunningham, P. Nuyujukian, V. Gilja, C. A. Chestek, S. I. Ryu, and K. V. Shenoy. A closed-loop human simulator for investigating the role of feedback-control in brain-machine interfaces. Journal of Neurophysiology, 105:1932-1949, 2011.

Abstract: Neural prosthetic systems seek to improve the lives of severely disabled people by decoding neural activity into useful behavioral commands. These systems and their decoding algorithms are typically developed "offline", using neural activity previously gathered from a healthy animal, and the decoded movement is then compared with the true movement that accompanied the recorded neural activity. However, this offline design and testing may neglect important features of a real prosthesis, most notably the critical role of feedback control, which enables the user to adjust neural activity while using the prosthesis. We hypothesize that under- standing and optimally designing high-performance decoders require an experimental platform where humans are in closed-loop with the various candidate decode systems and algorithms. It remains unexplored the extent to which the subject can, for a particular decode system, algorithm, or parameter, engage feedback and other strategies to improve decode performance. Closed-loop testing may suggest different choices than offline analyses. Here we ask if a healthy human subject, using a closed-loop neural prosthesis driven by synthetic neural activity, can inform system design. We use this online pros- thesis simulator (OPS) to optimize "online" decode performance based on a key parameter of a current state-of-the-art decode algorithm, the bin width of a Kalman filter. First, we show that offline and online analyses indeed suggest different parameter choices. Previous literature and our offline analyses agree that neural activity should be analyzed in bins of 100- to 300-ms width. OPS analysis, which incorporates feedback control, suggests that much shorter bin widths (25-50 ms) yield higher decode performance. Second, we confirm this surprising finding using a closed-loop rhesus monkey prosthetic system. These findings illustrate the type of discovery made possible by the OPS, and so we hypothesize that this novel testing approach will help in the design of prosthetic systems that will translate well to human patients.

Richard E. Turner and Maneesh Sahani. Demodulation as probabilistic inference. Transactions on Audio, Speech and Language Processing, 19:2398-2411, 2011.

Richard E. Turner and Maneesh Sahani. Probabilistic amplitude and frequency demodulation. In Advances in Neural Information Processing Systems 24, pages 981-989. The MIT Press, 2011.

Andrew Gordon Wilson and Zoubin Ghahramani. Generalised Wishart processes. In 27nd Conference on Uncertainty in Artificial Intelligence, 2011.

Comment: Supplementary Material, Best Student Paper Award

M. Zhao, A. P. Batista, J. P. Cunningham, C. A. Chestek, Z. Rivera-Alvidrez, R. Kalmar, S. I. Ryu, K. V. Shenoy, and S. Iyengar. An L1-regularized logistic model for detecting short-term neuronal interactions.. Journal of Computational Neuroscience, 2011, doi 10.1007/s10827-011-0365-5. In Press.

Abstract: Interactions among neurons are a key com- ponent of neural signal processing. Rich neural data sets potentially containing evidence of interactions can now be collected readily in the laboratory, but existing analysis methods are often not sufficiently sensitive and specific to reveal these interactions. Generalized linear models offer a platform for analyzing multi-electrode recordings of neuronal spike train data. Here we suggest an L1-regularized logistic regression model (L1L method) to detect short-term (order of 3ms) neuronal interactions. We estimate the parameters in this model using a coordinate descent algorithm, and determine the optimal tuning parameter using a Bayesian Information Criterion. Simulation studies show that in general the L1L method has better sensitivities and specificities than those of the traditional shuffle-corrected cross-correlogram (covariogram) method. The L1L method is able to detect excitatory interactions with both high sensitivity and specificity with reasonably large recordings, even when the magnitude of the interactions is small; similar results hold for inhibition given sufficiently high baseline firing rates. Our study also suggests that the false positives can be further removed by thresholding, because their magnitudes are typically smaller than true interactions. Simulations also show that the L1L method is somewhat robust to partially observed networks. We apply the method to multi-electrode recordings collected in the monkey dorsal premotor cortex (PMd) while the animal prepares to make reaching arm movements. The results show that some neurons interact differently depending on task conditions. The stronger interactions detected with our L1L method were also visible using the covariogram method.

Ryan Turner, Steven Bottone, and Zoubin Ghahramani. Fast online anomaly detection using scan statistics. In Samuel Kaski, David J. Miller, Erkki Oja, and Antti Honkela, editors, Machine Learning for Signal Processing (MLSP 2010), pages 385-390, Kittilä, Finland, August 2010.

Abstract: We present methods to do fast online anomaly detection using scan statistics. Scan statistics have long been used to detect statistically significant bursts of events. We extend the scan statistics framework to handle many practical issues that occur in application: dealing with an unknown background rate of events, allowing for slow natural changes in background frequency, the inverse problem of finding an unusual lack of events, and setting the test parameters to maximize power. We demonstrate its use on real and synthetic data sets with comparison to other methods.

Yunus Saatçi, Ryan Turner, and Carl Edward Rasmussen. Gaussian process change point models. In 27th International Conference on Machine Learning, pages 927-934, Haifa, Israel, June 2010.

Comment: poster, slides.

Comment: poster.

Richard E. Turner. Statistical Models for Natural Sounds. PhD thesis, Gatsby Computational Neuroscience Unit, UCL, 2010.

Andrew Gordon Wilson and Zoubin Ghahramani. Copula processes. In Advances in Neural Information Processing Systems 23, 2010. Spotlight.

Comment: Supplementary Material, slides.

Comment: poster.

Ryan Turner, Yunus Saatçi, and Carl Edward Rasmussen. Adaptive sequential Bayesian change point detection. In Zaïd Harchaoui, editor, NIPS Workshop on Temporal Segmentation, Whistler, BC, Canada, December 2009.

Abstract: Real-world time series are often nonstationary with respect to the parameters of some underlying prediction model (UPM). Furthermore, it is often desirable to adapt the UPM to incoming regime changes as soon as possible, necessitating sequential inference about change point locations. A Bayesian algorithm for online change point detection (BOCPD) has been introduced recently by Adams and MacKay (2007). In this algorithm, uncertainty about the last change point location is updated sequentially, and is integrated out to make online predictions robust to parameter changes. BOCPD requires a set of fixed hyper-parameters which allow the user to fully specify the hazard function for change points and the prior distribution over the parameters of the UPM. In practice, finding the "right" hyper-parameters can be quite difficult. We therefore extend BOCPD by introducing hyper-parameter learning, without sacrificing the online nature of the algorithm. Hyper-parameter learning is performed by optimizing the marginal likelihood of the BOCPD model, a closed-form quantity which can be computed sequentially. We illustrate performance on three real-world datasets.

Comment: poster, slides.

Comment: With corrections. code.

Richard E. Turner and Maneesh Sahani. Modeling natural sounds with modulation cascade processes. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, nips20, volume 20. mit, 2008.

Richard E. Turner and M Sahani. Probabilistic amplitude demodulation. In 7th International Conference on Independent Component Analysis and Signal Separation, pages 544-551, 2007.

Richard E. Turner and Maneesh Sahani. A maximum-likelihood interpretation for slow feature analysis. nc, 19(4):1022-1038, 2007, doi http://dx.doi.org/10.1162/neco.2007.19.4.1022.

Abstract: The brain extracts useful features from a maelstrom of sensory information, and a fundamental goal of theoretical neuroscience is to work out how it does so. One proposed feature extraction strategy is motivated by the observation that the meaning of sensory data, such as the identity of a moving visual object, is often more persistent than the activation of any single sensory receptor. This notion is embodied in the slow feature analysis (SFA) algorithm, which uses “slowness” as an heuristic by which to extract semantic information from multi-dimensional time-series. Here, we develop a probabilistic interpretation of this algorithm showing that inference and learning in the limiting case of a suitable probabilistic model yield exactly the results of SFA. Similar equivalences have proved useful in interpreting and extending comparable algorithms such as independent component analysis. For SFA, we use the equivalent probabilistic model as a conceptual spring-board, with which to motivate several novel extensions to the algorithm.

Comment: techreport

Pedro A. d. F. R. Højen-Sørensen, Carl Edward Rasmussen, and Lars Kai Hansen. Bayesian modelling of fMRI time series. In Advances in Neural Information Processing Systems 12, pages 754-760. The MIT Press, 2000.

Abstract: We present a Hidden Markov Model (HMM) for inferring the hidden psychological state (or neural activity) during single trial fMRI activation experiments with blocked task paradigms. Inference is based on Bayesian methodology, using a combination of analytical and a variety of Markov Chain Monte Carlo (MCMC) sampling techniques. The advantage of this method is that detection of short time learning effects between repeated trials is possible since inference is based only on single trial experiments.

Network Modelling

Novi Quadrianto, Chao Chen, and Christoph Lampert. The most persistent soft-clique in a set of sampled graphs. In 29th International Conference on Machine Learning, Edinburgh, Scotland, June 2012.

Abstract: When searching for characteristic subpatterns in potentially noisy graph data, it appears self-evident that having multiple observations would be better than having just one. However, it turns out that the inconsistencies introduced when different graph instances have different edge sets pose a serious challenge. In this work we address this challenge for the problem of finding maximum weighted cliques. We introduce the concept of most persistent soft-clique. This is subset of vertices, that 1) is almost fully or at least densely connected, 2) occurs in all or almost all graph instances, and 3) has the maximum weight. We present a measure of clique-ness, that essentially counts the number of edge missing to make a subset of vertices into a clique. With this measure, we show that the problem of finding the most persistent soft-clique problem can be cast either as: a) a max-min two person game optimization problem, or b) a min-min soft margin optimization problem. Both formulations lead to the same solution when using a partial Lagrangian method to solve the optimization problems. By experiments on synthetic data and on real social network data we show that the proposed method is able to reliably find soft cliques in graph data, even if that is distorted by random noise or unreliable observations.

Active Learning

Tomoharu Iwata, Neil Houlsby, and Zoubin Ghahramani. Active learning for interactive visualization. In 16th International Conference on Artificial Intelligence and Statistics, 2013.

Abstract: Many automatic visualization methods have been proposed. However, a visualization that is automatically generated might be different to how a user wants to arrange the objects in visualization space. By allowing users to re-locate objects in the embedding space of the visualization, they can adjust the visualization to their preference. We propose an active learning framework for interactive visualization which selects objects for the user to re-locate so that they can obtain their desired visualization by re-locating as few as possible. The framework is based on an information theoretic criterion, which favors objects that reduce the uncertainty of the visualization. We present a concrete application of the proposed framework to the Laplacian eigenmap visualization method. We demonstrate experimentally that the proposed framework yields the desired visualization with fewer user interactions than existing methods.

Konstantin Kravtsov, Stanislav Straupe, Igor Radchenko, Neil Houlsby, Ferenc Huszár, and Sergey Kulik. Experimental adaptive Bayesian tomography. Physical Review A, 87(6):062122, 2013.

Abstract: We report an experimental realization of an adaptive quantum state tomography protocol. Our method takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states we observe close to $N^-1$ scaling of infidelity with overall number of registered events, while best non-adaptive protocols allow for $N^-1/2$ scaling only. Experiments are performed for polarization qubits, but the approach is readily adapted to any dimension.

Ferenc Huszár and Neil Houlsby. Adaptive Bayesian quantum tomography. Physical Review A, 85(5):052120, 2012.

Abstract: In this paper we revisit the problem of optimal design of quantum tomographic experiments. In contrast to previous approaches where an optimal set of measurements is decided in advance of the experiment, we allow for measurements to be adaptively and efficiently re-optimised depending on data collected so far. We develop an adaptive statistical framework based on Bayesian inference and Shannon's information, and demonstrate a ten-fold reduction in the total number of measurements required as compared to non-adaptive methods, including mutually unbiased bases.

Neuroscience

Neil Houlsby, Ferenc Huszár, Mohammad M Ghassemi, Gergő Orbán, Daniel M Wolpert, and Máté Lengyel. Cognitive tomography reveals complex task-independent mental representations. Current Biology, 23(21):2169-2175, 2013, doi 10.1016/j.cub.2013.09.012.

Abstract: Humans develop rich mental representations that guide their behavior in a variety of every-day tasks. However, it is unknown whether these representations, often formalized as priors in Bayesian inference, are specific for each task or subserve multiple tasks. Current approaches cannot distinguish between these two possibilities because they cannot extract comparable representations across different tasks. Here, we develop a novel method, termed cognitive tomography, that can extract complex, multi-dimensional priors across tasks. We apply this method to human judgments in two qualitatively different tasks, familiarity and odd-one-out, involving an ecologically relevant set of stimuli, human faces. We show that priors over faces are structurally complex and vary dramatically across subjects, but are invariant across the tasks within each subject. The priors we extract from each task allow us to predict with high precision the behavior of subjects for novel stimuli both in the same task as well as in the other task. Our results provide the first evidence for a single high-dimensional structured representation of a naturalistic stimulus set that guides behavior in multiple tasks. Moreover, the representations estimated by cognitive tomography can provide independent, behavior-based regressors for elucidating the neural correlates of complex naturalistic priors.

M. M. Churchland, J. P. Cunningham, M. T. Kaufman, S. I. Ryu, and K. V. Shenoy. Cortical preparatory activity: Representation of movement or first cog in a dynamical machine?. Neuron, 68:387-400, 2010.

Abstract: The motor cortices are active during both movement and movement preparation. A common assumption is that preparatory activity constitutes a subthreshold form of movement activity: a neuron active during rightward movements becomes modestly active during preparation of a rightward movement. We asked whether this pattern of activity is, in fact, observed. We found that it was not: at the level of a single neuron, preparatory tuning was weakly correlated with movement-period tuning. Yet, somewhat paradoxically, preparatory tuning could be captured by a preferred direction in an abstract "space" that described the population-level pattern of movement activity. In fact, this relationship accounted for preparatory responses better than did traditional tuning models. These results are expected if preparatory activity provides the initial state of a dynamical system whose evolution produces movement activity. Our results thus suggest that preparatory activity may not represent specific factors, and may instead play a more mechanistic role.

Richard E. Turner. Statistical Models for Natural Sounds. PhD thesis, Gatsby Computational Neuroscience Unit, UCL, 2010.

C. Chang, J. P. Cunningham, and G. Glover. Influence of heart rate on the bold signal: the cardiac response function. NeuroImage, 44:857-869, 2009.

Comment: Spotlight Presentation

J. P. Cunningham. Derivation of Expectation Propagation for "fast Gaussian process methods for point process intensity estimation". Technical report, Stanford University, 2008.

Richard E. Turner and Maneesh Sahani. A maximum-likelihood interpretation for slow feature analysis. nc, 19(4):1022-1038, 2007, doi http://dx.doi.org/10.1162/neco.2007.19.4.1022.

Signal Processing

Richard E. Turner and Maneesh Sahani. Demodulation as probabilistic inference. Transactions on Audio, Speech and Language Processing, 19:2398-2411, 2011.

Richard E. Turner and Maneesh Sahani. Probabilistic amplitude and frequency demodulation. In Advances in Neural Information Processing Systems 24, pages 981-989. The MIT Press, 2011.

Richard E. Turner. Statistical Models for Natural Sounds. PhD thesis, Gatsby Computational Neuroscience Unit, UCL, 2010.

Richard E. Turner and Maneesh Sahani. Modeling natural sounds with modulation cascade processes. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, nips20, volume 20. mit, 2008.

Richard E. Turner and M Sahani. Probabilistic amplitude demodulation. In 7th International Conference on Independent Component Analysis and Signal Separation, pages 544-551, 2007.

Machine Vision

Viktoriia Sharmanska, Novi Quadrianto, and Christoph Lampert. Learning to rank using privileged information. In International Conference on Computer Vision, 2013.

Abstract: Many computer vision problems have an asymmetric distribution of information between training and test time. In this work, we study the case where we are given additional information about the training data, which however will not be available at test time. This situation is called learning using privileged information (LUPI). We introduce two maximum-margin techniques that are able to make use of this additional source of information, and we show that the framework is applicable to several scenarios that have been studied in computer vision before. Experiments with attributes, bounding boxes, image tags and rationales as additional information in object classification show promising results.

Viktoriia Sharmanska, Novi Quadrianto, and Christoph Lampert. Augmented attributes representations. In 12th European Conference on Computer Vision, pages 242-255, 2012.

Abstract: We propose a new learning method to infer a mid-level feature representation that combines the advantage of semantic attribute representations with the higher expressive power of non-semantic features. The idea lies in augmenting an existing attribute-based representation with additional dimensions for which an autoencoder model is coupled with a large-margin principle. This construction allows a smooth transition between the zero-shot regime with no training example, the unsupervised regime with training examples but without class labels, and the supervised regime with training examples and with class labels. The resulting optimization problem can be solved efficiently, because several of the necessity steps have closed-form solutions. Through extensive experiments we show that the augmented representation achieves better results in terms of object categorization accuracy than the semantic representation alone.

Tatiana Tommasi, Novi Quadrianto, Barbara Caputo, and Christoph Lampert. Beyond dataset bias: Multi-task unaligned shared knowledge transfer. In 11th Asian Conference on Computer Vision, 2012.

Abstract: Many visual datasets are traditionally used to analyze the performance of different learning techniques. The evaluation is usually done within each dataset, therefore it is questionable if such results are a reliable indicator of true generalization ability. We propose here an algorithm to exploit the existing data resources when learning on a new multiclass problem. Our main idea is to identify an image representation that decomposes orthogonally into two subspaces: a part specific to each dataset, and a part generic to, and therefore shared between, all the considered source sets. This allows us to use the generic representation as un-biased reference knowledge for a novel classification task. By casting the method in the multi-view setting, we also make it possible to use different features for different databases. We call the algorithm MUST, Multitask Unaligned Shared knowledge Transfer. Through extensive experiments on five public datasets, we show that MUST consistently improves the cross-datasets generalization performance.

Richard E. Turner and Maneesh Sahani. A maximum-likelihood interpretation for slow feature analysis. nc, 19(4):1022-1038, 2007, doi http://dx.doi.org/10.1162/neco.2007.19.4.1022.

Machine Hearing

Richard E. Turner and Maneesh Sahani. Demodulation as probabilistic inference. Transactions on Audio, Speech and Language Processing, 19:2398-2411, 2011.

Richard E. Turner and Maneesh Sahani. Probabilistic amplitude and frequency demodulation. In Advances in Neural Information Processing Systems 24, pages 981-989. The MIT Press, 2011.

Richard E. Turner. Statistical Models for Natural Sounds. PhD thesis, Gatsby Computational Neuroscience Unit, UCL, 2010.

Richard E. Turner and Maneesh Sahani. Modeling natural sounds with modulation cascade processes. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, nips20, volume 20. mit, 2008.

Richard E. Turner and M Sahani. Probabilistic amplitude demodulation. In 7th International Conference on Independent Component Analysis and Signal Separation, pages 544-551, 2007.

Review Articles and Tutorials

Zoubin Ghahramani. Bayesian nonparametrics and the probabilistic approach to modelling. Philosophical Transactions of the Royal Society A, 2012.

Thomas L. Griffiths and Zoubin Ghahramani. The Indian buffet process: An introduction and review. Journal of Machine Learning Research, 12:1185-1224, April 2011.

Sören Sonnenburg, Mikio L. Braun, Cheng Soon Ong, Samy Bengio, Leon Bottou, Geoffrey Holmes, Yann LeCun, Klaus-Robert Müller, Fernando Pereira, Carl Edward Rasmussen, Gunnar Rätsch, Bernhard Schölkopf, Alexander Smola, Pascal Vincent, Jason Weston, and Robert C. Williamson. The need for open source software in machine learning. Journal of Machine Learning Research, 8:2443-2466, October 2007.

Abstract: Open source tools have recently reached a level of maturity which makes them suitable for building large-scale real-world systems. At the same time, the field of machine learning has developed a large body of powerful learning algorithms for diverse applications. However, the true potential of these methods is not realized, since existing implementations are not openly shared, resulting in software with low usability, and weak interoperability. We argue that this situation can be significantly improved by increasing incentives for researchers to publish their software under an open source model. Additionally, we outline the problems authors are faced with when trying to publish algorithmic implementations of machine learning methods. We believe that a resource of peer reviewed software accompanied by short articles would be highly valuable to both the machine learning and the general scientific community.

Comment: Copyright by Springer, springerlink

Carl Edward Rasmussen and Zoubin Ghahramani. Occam's razor. In Advances in Neural Information Processing Systems 13, pages 294-300, Cambridge, MA, USA, December 2001. The MIT Press.

Abstract: The Bayesian paradigm apparently only sometimes gives rise to Occam's Razor; at other times very large models perform well. We give simple examples of both kinds of behaviour. The two views are reconciled when measuring complexity of functions, rather than of the machinery used to implement them. We analyze the complexity of functions for some linear in the parameter models that are equivalent to Gaussian Processes, and always find Occam's Razor at work.