Inference by Variable Elimination

Abstract

We present in this chapter one of the simplest methods for general inference in Bayesian networks, which is based on the principle of variable elimination: A process by which we successively remove variables from a Bayesian network while maintaining its ability to answer queries of interest. Introduction We saw in Chapter 5 how a number of real-world problems can be solved by posing queries with respect to Bayesian networks. We also identified four types of queries: probability of evidence, prior and posterior marginals, most probable explanation (MPE), and maximum a posterior hypothesis (MAP). We present in this chapter one of the simplest inference algorithms for answering these types of queries, which is based on the principle of variable elimination. Our interest here will be restricted to computing the probability of evidence and marginal distributions, leaving the discussion of MPE and MAP queries to Chapter 10. We start in Section 6.2 by introducing the process of eliminating a variable. This process relies on some basic operations on a class of functions known as factors, which we discuss in Section 6.3. We then introduce the variable elimination algorithm in Section 6.4 and see how it can be used to compute prior marginals in Section 6.5. The performance of variable elimination will critically depend on the order in which we eliminate variables. We discuss this issue in Section 6.6, where we also provide some heuristics for choosing good elimination orders.