Stat 700
Probabilistic graphical models

Announcements

• Nov 13: Homework 4 is issued.
• Oct 26: Homework 3 is issued.
• Oct 15: Partial solutions to homework1 is posted in Ctools.
• Homework 2 is out on Oct 6.
• A typo in problem 6(c) (Homework 1): The cost of the computation of $p(x_i)$ is (at most) proportional to the number of edges in $G$.
• Office hour changed to Weds 10-11am, or by appointment.
• Homework 1 is out on Thurs Sept 17.

Topics

• Basics on graphical models, Markov properties, recursive decomposability, elimination algorithms
• Sum-product algorithm, factor graphs, semi-rings
• Frequentist and Bayesian methods
• Bayesian classification, linear models and generalized linear (GLIM) models, on-line methods
• Exponential family, sufficiency, conjugacy
• Legendre-Fenchel duality, information theoretic concepts
• The EM algorithm
• Mixture models, conditional mixture models, hierarchical mixture models
• Hidden Markov models (HMM) and forward-backward algorithms
• Factor analysis, principal component analysis (PCA), canonical correlation analysis (CCA), independent component analysis (ICA)
• Kalman filtering and Rauch-Tung-Striebel smoothing
• Markov properties of graphical models
• Junction tree algorithm
• Chains, trees, factorial models, coupled models, layered models
• Importance sampling, Markov Chain Monte Carlo methods (Gibbs/ Metropolis-Hastings samplers)
• Variational inference algorithms: mean field, belief propagation, convex relaxations
• Model selection, marginal likelihood, AIC, BIC and MDL
• Nonparametric methods: Gaussian processes, Dirichlet processes, infinite mixture models
• Applications to signal processing, bioinformatics, communication, natural language processing, computer vision etc.

Prerequisites

Text books

• M. I. Jordan, An introduction to probabilistic graphical models. This book is not yet published. Hard copies of selected book chapters will be distributed in a classpack from the Dollar Bill Copying on Church Street.

• S. Lauritzen, Graphical models, Publisher: Oxford University Press, 1996.
• J. Whittaker, Graphical Models in Applied Multivariate Statistics, Wiley, 2009.
• N. Cressie, Statistics for Spatial Data, Wiley, NY, 1993.
• S. Banerjee, B.P. Carlin and A.E. Gelfand, Hierarchical Modeling and Analysis for Spatial Data, Chapman and Hall/CRC Press, 2004.

Tentative Schedule and Reading Assignments

• 9/10: Intro to GM's and exact inference: GM with directed graphs, undirected graphs, factor graphs; Elimination algorithms, probability propagation algorithms
  • Graphical models. M. I. Jordan. Statistical Science (Special Issue on Bayesian Statistics), 19, 140-155, 2004. Section 3 in this overview article can be skipped at the first reading.
  • Read MJ's Chapter 2.

• 9/17: Continue on exact inference (sum-product, max-product algorithms), basic statistical concepts (frequentist and Bayesian estimation methods)
  • Chapters 3/4, and beginning of Chapter 5
  • Hw 1 out, due 9/24

• 9/24: Canonical estimation problems from the viewpoint of graphical models (density estimation, linear regression and classification; supervised vs unsupervised learning; generative vs discriminative models )
  • Chapters 5, 6 and 7

• 10/1: Exponential family, convexity and Legendre-Fenchel duality, sufficiency and some information-theoretic concepts, conjugate priors
  • Chapter 8, and Section 3 of the Wainwright-Jordan paper (see the link below)

• 10/8: Continue on exponential families

• 10/15: Mixture models and estimation(Chap 9/10/11): Completely observed GMs, mixtures and conditional mixtures, the EM algorithm

• 10/22: Continue on mixture models and the EM algorithm

• 10/29: Hidden Markov models (Chap 12), Multivariate Gaussians (Chap 13)

• 11/5: Factor analysis (Chap 14), Kalman filter and smoothing (Chap 15)

• 11/12: Approximate inference: Sampling methods (important sampling, Markov chain Monte Carlo)
  • Chapter 21
  • X. Meng and W. Wong, Simulating ratios of normalizing constants via a simple identity: A theoretical exploration. Statistica Sinica 6(1996), 831-860.
  • A. Gelman and X. Meng, Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Statist. Sci. Volume 13, Number 2 (1998), 163-185.
  • M. Jerrum and A. Sinclair, Polynomial-time Approximation Algorithms for the Ising Model, SIAM Journal on Computing 22 (1993), pp. 1087-1116.

• 11/19: The junction tree algorithm (Chap 17), Approximate inference: variational inference methods for Markov random fields and directed GM's
  • Read Chapter 3 and 5 of Wainwright-Jordan's.
  • Graphical models, exponential families, and variational inference. M. J. Wainwright and M. I. Jordan. Foundations and Trends in Machine Learning, 1, 1-305, 2008.

• 11/26: Thanksgiving break

• 12/3: Variational inference (cont.); Model selection (Chap 26)
  

• 12/10: Model selection methods:
  • Frequentist approaches using penalties on model complexity
  • Bayesian approaches via the marginal likelihood
  • Nonparametric Bayesian methods using random measures/ stochastic processes as the prior on models

Homeworks

• Homework 1 (out Sep 17, due Sep 24) [ pdf ]
• Homework 2 (out Oct 6, due Oct 15) [ pdf ]
• Homework 3 (out Oct 26, due Nov 5) [ pdf ]
• Homework 4 (out Nov 13, due Dec 3) [ pdf ]

Projects

Structure/Evaluation