State estimation in discrete graphical models

p(X1:D|G, θ) (1) whereG is the graph structure (either directed or undirected or both), and θ are the parameters. In Bayesian modeling, we treat the parameters as random variables as well, but they are in turn conditioned on fixed hyper parameters α: p(X1:D, θ|G,α) (2) Clearly this can be represented as in Equation 1 by appropriately redefining X and θ. It will also be notationally helpful to distinguish the hidden nodes X from the observed nodes Y . Without loss of generality, we may assume there is a Yi node for every Xi node; we sometimes call Yi the local evidence for Xi. If Xi is not observed, then Yi can be set to something non informative. There are many quantities of interest we may be interested in inferring. Broadly speaking, they are as follows 1. State estimation: inferring p(X |y, θ,G). 2. Parameter estimation (learning): inferring p(θ|y,G). 3. Model selection (structure learning): inferring p(G|y). In this chapter, we focus on the first problem. Furthermore, we restrict our attention to the case where the Xi are discrete random variables. (The observed yi’s may be continuous, however.) The techniques we describe work equally well for directed and undirected models. A directed model