## Publications

#### Beta diffusion trees

Creighton Heaukulani, David A. Knowles, Zoubin Ghahramani, June 2014. (In 31st International Conference on Machine Learning). Beijing, China.

Abstract▼ URL

We define the beta diffusion tree, a random tree structure with a set of leaves that defines a collection of overlapping subsets of objects, known as a feature allocation. The generative process for the tree is defined in terms of particles (representing the objects) diffusing in some continuous space, analogously to the Dirichlet and Pitman–Yor diffusion trees (Neal, 2003b; Knowles & Ghahramani, 2011), both of which define tree structures over clusters of the particles. With the beta diffusion tree, however, multiple copies of a particle may exist and diffuse to multiple locations in the continuous space, resulting in (a random number of) possibly overlapping clusters of the objects. We demonstrate how to build a hierarchically-clustered factor analysis model with the beta diffusion tree and how to perform inference over the random tree structures with a Markov chain Monte Carlo algorithm. We conclude with several numerical experiments on missing data problems with data sets of gene expression arrays, international development statistics, and intranational socioeconomic measurements.

#### Beta diffusion trees and hierarchical feature allocations

Creighton Heaukulani, David A. Knowles, Zoubin Ghahramani, August 2014. Dept. of Engineering, University of Cambridge,

Abstract▼ URL

We define the beta diffusion tree, a random tree structure with a set of leaves that defines a collection of overlapping subsets of objects, known as a feature allocation. A generative process for the tree structure is defined in terms of particles (representing the objects) diffusing in some continuous space, analogously to the Dirichlet diffusion tree (Neal, 2003b), which defines a tree structure over partitions (i.e., non-overlapping subsets) of the objects. Unlike in the Dirichlet diffusion tree, multiple copies of a particle may exist and diffuse along multiple branches in the beta diffusion tree, and an object may therefore belong to multiple subsets of particles. We demonstrate how to build a hierarchically-clustered factor analysis model with the beta diffusion tree and how to perform inference over the random tree structures with a Markov chain Monte Carlo algorithm. We conclude with several numerical experiments on missing data problems with data sets of gene expression microarrays, international development statistics, and intranational socioeconomic measurements.

#### The combinatorial structure of beta negative binomial processes

Creighton Heaukulani, Daniel M. Roy, March 2014. Dept. of Engineering, University of Cambridge,

Abstract▼ URL

We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for unknown multisets of a measurable space. Previous work has characterized random subsets arising from conditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure. In this case, the combinatorial structure is described by the Indian buffet process. Our results give a count analogue of the Indian buffet process, which we call a negative binomial Indian buffet process. As an intermediate step toward this goal, we provide constructions for the beta negative binomial process that avoid a representation of the underlying beta process base measure.