Probabilistic Machine Learning chunks

The entire course falls naturally in three parts, Gaussian processes, probabilistic ranking and text modeling.

Part I: Supervised non-parametric probabilistic inference using Gaussian processes

In a nutshell, part I is concerned with...

1. Modelling data
   • goals of building a model
   • requirements for good models
   • data, parameters and latent variables
2. Linear in the parameters regression
   • making predictions, concept of a model
   • least squares fit, and the Normal equations
   • requires linear algebra
   • model complexity: underfitting and overfitting
3. Likelihood and the concept of noise
   • Gaussian independent and identically distributed (iid) noise
   • Maximum likelihood fitting
   • Equivalence to least squares
   • Motivation of inference with multiple hypothesis
4. Probability fundamentals
   • Medical example
   • Joint, conditional and marginal probabilities
   • The two rules of probability: sum and product
   • Bayes' rule
5. Bayesian inference and prediction with finite regression models
   • Likelihood and prior
   • Posterior and predictive distribution, with algebra and pictorially
   • the marginal likelihood
6. Background: Some useful Gaussian and Matrix equations
   • matrix inversion lemma
   • mean and variance of Gaussian
   • mean and variance of projection of Gaussian
   • marginal and conditional of Gaussian
   • products of Gaussians
7. Marginal likelihood
   • Bayesian model selection
   • MCMC based explanation of how the marginal likelihood works
   • Average the likelihood over the prior: example
8. Distributions over parameters and over functions
   • Concept of prior over functions and over parameters
   • nuissance parameters
   • Could we sidestep parameters, and work directly with functions?
9. Gaussian process
   • From scalar Gaussians to multivariate Gaussians to Gaussian processes
   • Functions are like infinitely long vectors, GPs are distributions over functions
   • Marginal and conditional Gaussian
   • GP definition
   • Conditional generation and joint generation
10. Gaussian processes and data
    • In pictures: prior and posterior
    • In algebra: prior and posterior
    • An analytic marginal likelihood, and some intuition
11. Gaussian process marginal likelihood and hyperparameters
    • the GP marginal likelihood, and it interpretation
    • hyperparameters can control the properties of functions
    • example: finding hyperparameters by maximizing the marginal likelihood
    • Occam's Razor
12. Correspondence between Linear in the parameters models and GPs
• From linear in the parameters models to GPs
• From GPs to linear in the parameters models
• Computational considerations: which is more efficient?
13. covariance functions
    • Stationary covariance functions, squared exponential, rational quadratic, Matérn covariance function
    • periodic covariance
    • neural network covariance function
    • Combining simple covariance functions into more interesting ones
14. The gpml toolbox

Part II: Ranking

15. Ranking: motivation and tennis example
    • Competition in sports and games (TrueSkill problem, match making)
    • Tennis: the ATP ranking system explained
    • Shortcomings: what does one need to make actual predictions? (who wins?)
    • The TrueSkill ranking model
16. Gibbs sampling
    • Calulating integrals using sampling
    • Markov chains and invariant distributions
    • Gibbs sampling
17. Gibbs sampling in TrueSkill
    • Conditional distributions in TrueSkill are tractable
18. Representing distributions using factor graphs
    • the cost of computing marginal distributions
    • algebraic and graphical representations
    • local computations on the graph
    • message passing: the sum-product rules
19. message passing in TrueSkill
    • messages are not all tractable
20. Approximate messages using moment matching
    • How to approximate a step function by a Gaussian?

Part III: Modeling text

33. Modeling text
• Modeling collections of documents
• probabilistic models of text
• Bag of words models
• Zipf's law
34. Discrete distributions on binary variables (tossing coins)
    • Binary variables and the Bernoulli distribution
    • Sequences, the binomial and discrete distributions
    • Inference and the Beta distributinos: probabilities over probabilities
35. Discrete distributions over multiple outcomes
• multinomials, categorical and discrete distributions
• inference and the Dirichlet prior
36. Document models
• Categorical model
• Mixture of categoricals model
• Trainig mixture models with EM
• A Bayesian mixture model
37. The Expectation Maximization (EM) algoritm
• Maximum likelihood in models with latent variables
38. Gibbs sampling for Bayesian mixture model
• Gibbs sampling
• Collapsed Gibbs sampling
39. Latent Dichichlet Allocation topic models
• A more interesting topic model
• Inference using Gibbs sampling