Cristopher M. Bishop's tutorial on graphical models
- 1. Part 1: Graphical Models Machine Learning Techniques for Computer Vision Microsoft Research Cambridge ECCV 2004, Prague Christopher M. Bishop
- 2. About this Tutorial Learning is the new frontier in computer vision Focus on concepts not lists of algorithms not technical details Graduate level Please ask questions!
- 3. Overview Part 1: Graphical models directed and undirected graphs inference and learning Part 2: Unsupervised learning mixture models, EM variational inference, model complexity continuous latent variables Part 3: Supervised learning decision theory linear models, neural networks, boosting, sparse kernel machines
- 4. Probability Theory Sum rule Product rule From these we have Bayes’ theorem with normalization
- 5. Role of the Graphs New insights into existing models Motivation for new models Graph based algorithms for calculation and computation c.f. Feynman diagrams in physics
- 6. Decomposition Consider an arbitrary joint distribution By successive application of the product rule
- 7. Directed Acyclic Graphs Joint distribution where denotes the parents of i No directed cycles
- 8. Undirected Graphs Provided then joint distribution is product of non-negative functions over the cliques of the graph where are the clique potentials, and Z is a normalization constant
- 9. Conditioning on Evidence Variables may be hidden (latent) or visible (observed) Latent variables may have a specific interpretation, or may be introduced to permit a richer class of distribution
- 10. Conditional Independences x independent of y given z if, for all values of z , For undirected graphs this is given by graph separation!
- 11. “Explaining Away” C.I. for directed graphs similar, but with one subtlety Illustration: pixel colour in an image image colour surface colour lighting colour
- 13. Example: State Space Models Hidden Markov model Kalman filter
- 15. Example: Factorial SSM Multiple hidden sequences Avoid exponentially large hidden space
- 16. Example: Markov Random Field Typical application: image region labelling
- 17. Example: Conditional Random Field
- 18. Inference Simple example: Bayes’ theorem
- 19. Message Passing Example Find marginal for a particular node for M -state nodes, cost is exponential in length of chain but, we can exploit the graphical structure (conditional independences)
- 20. Message Passing Joint distribution Exchange sums and products
- 21. Message Passing Express as product of messages Recursive evaluation of messages Find Z by normalizing
- 22. Belief Propagation Extension to general tree-structured graphs At each node: form product of incoming messages and local evidence marginalize to give outgoing message one message in each direction across every link Fails if there are loops
- 23. Junction Tree Algorithm An efficient exact algorithm for a general graph applies to both directed and undirected graphs compile original graph into a tree of cliques then perform message passing on this tree Problem: cost is exponential in size of largest clique many vision models have intractably large cliques
- 24. Loopy Belief Propagation Apply belief propagation directly to general graph need to keep iterating might not converge State-of-the-art performance in error-correcting codes
- 25. Max-product Algorithm Goal: find define then Message passing algorithm with “sum” replaced by “max” Example: Viterbi algorithm for HMMs
- 26. Inference and Learning Data set Likelihood function (independent observations) Maximize (log) likelihood Predictive distribution
- 27. Regularized Maximum Likelihood Prior , posterior MAP (maximum posterior) Predictive distribution Not really Bayesian
- 28. Bayesian Learning Key idea is to marginalize over unknown parameters, rather than make point estimates avoids severe over-fitting of ML and MAP allows direct model comparison Parameters are now latent variables Bayesian learning is an inference problem!
- 31. And Finally … the Exponential Family Many distributions can be written in the form Includes: Gaussian Dirichlet Gamma Multi-nomial Wishart Bernoulli … Building blocks in graphs to give rich probabilistic models
- 32. Illustration: the Gaussian Use precision (inverse variance) In standard form
- 33. Maximum Likelihood Likelihood function (independent observations) Depends on data via sufficient statistics of fixed dimension
- 34. Conjugate Priors Prior has same functional form as likelihood Hence posterior is of the form Can interpret prior as effective observations of value Examples: Gaussian for the mean of a Gaussian Gaussian-Wishart for mean and precision of Gaussian Dirichlet for the parameters of a discrete distribution
- 35. Summary of Part 1 Directed graphs Undirected graphs Inference by message passing: belief propagation