A Variational Principle for Graphical Models

Graphical models bring together graph theory and probability theory in a powerful formalism for multivariate statistical modeling. In statistical signal processing— as well as in related fields such as communication theory, control theory and bioinformatics—statistical models have long been formulated in terms of graphs, and algorithms for computing basic statistical quantities such as likelihoods and marginal probabilities have often been expressed in terms of recursions operating on these graphs. Examples include hidden Markov models, Markov random fields, the forward-backward algorithm and Kalman filtering [ Rabiner and Juang (1993); Pearl (1988); Kailath et al. (2000)]. These ideas can be understood, unified and generalized within the formalism of graphical models. Indeed, graphical models provide a natural framework for formulating variations on these classical architectures, and for exploring entirely new families of statistical models. The recursive algorithms cited above are all instances of a general recursive algorithm known as the junction tree algorithm [ Lauritzen and Spiegelhalter, 1988]. The junction tree algorithm takes advantage of factorization properties of the joint probability distribution that are encoded by the pattern of missing edges in a graphical model. For suitably sparse graphs, the junction tree algorithm provides a systematic and practical solution to the general problem of computing likelihoods and other statistical quantities associated with a graphical model. Unfortunately, many graphical models of practical interest are not " suitably sparse, " so that the junction tree algorithm no longer provides a viable computational solution to the problem of computing marginal probabilities and other expectations. One popular source of methods for attempting to cope with such cases is the Markov chain Monte Carlo (MCMC) framework, and indeed there is a significant literature on