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.

We discuss in this chapter a class of approximate inference algorithms which are based on belief propagation. These algorithms provide a full spectrum of approximations, allowing one to trade-off approximation quality with computational resources.

Introduction

The algorithm of belief propagation was first introduced as a specialized algorithm that applied only to networks having a polytree structure. This algorithm, which we treated in Section 7.5.4, was later applied to networks with arbitrary structure and found to produce high-quality approximations in certain cases. This observation triggered a line of investigations into the semantics of belief propagation, which had the effect of introducing a generalization of the algorithm that provides a full spectrum of approximations with belief propagation approximations at one end and exact results at the other.

We discuss belief propagation as applied to polytrees in Section 14.2 and then discuss its application to more general networks in Section 14.3. The semantics of belief propagation are exposed in Section 14.4, showing howit can be viewed as searching for an approximate distribution that satisfies some interesting properties. These semantics will then be the basis for developing generalized belief propagation in Sections 14.5–14.7. An alternative semantics for belief propagation will also be given in Section 14.8, together with a corresponding generalization. The difference between the two generalizations of belief propagation is not only in their semantics but also in the way they allow the user to trade off the approximation quality with the computational resources needed to produce them.

We consider in this chapter the computational complexity of probabilistic inference. We also provide some reductions of probabilistic inference to well known problems, allowing us to benefit from specialized algorithms that have been developed for these problems.

Introduction

In previous chapters, we discussed algorithms for answering three types of queries with respect to a Bayesian network that induces a distribution Pr(X). In particular, given some evidence e we discussed algorithms for computing:

• The probability of evidence e, Pr(e) (see Chapters 6–8)

• The MPE probability for evidence e, MPEP (e) (see Chapter 10)

• The MAP probability for variables Q and evidence e, MAPP (Q, e) (see Chapter 10).

In this chapter, we consider the complexity of three decision problems that correspond to these queries. In particular, given a number p, we consider the following problems:

• D-PR: Is Pr(e) > p?

• D-MPE: Is there a network instantiation x such that Pr(x, e) > p?

• D-MAP: Given variables Q ⊆ X, is there an instantiation q such that Pr(q, e) > p?

We also consider a fourth decision problem that includes D-PR as a special case:

• D-MAR: Given variables Q ⊆ X and instantiation q, is Pr(q|e) > p?

Note here that when e is the trivial instantiation, D-MAR reduces to asking whether Pr(q) > p, which is identical to D-PR.

We provide a number of results on these decision problems in this chapter. In particular, we show in Sections 11.2–11.4 that D-MPE is NP-complete, D-PR and D-MAR are PPcomplete, and D-MAP is NPPP-complete.

We discuss in this chapter a class of approximate inference algorithms based on stochastic sampling: a process by which we repeatedly simulate situations according to their probability and then estimate the probabilities of events based on the frequency of their occurrence in the simulated situations.

Introduction

Consider the Bayesian network in Figure 15.1 and suppose that our goal is to estimate the probability of some event, say, wet grass. Stochastic sampling is a method for estimating such probabilities that works by measuring the frequency at which events materialize in a sequence of situations simulated according to their probability of occurrence. For example, if we simulate 100 situations and find out that the grass is wet in 30 of them, we estimate the probability of wet grass to be 3/10. As we see later, we can efficiently simulate situations according to their probability of occurrence by operating on the corresponding Bayesian network, a process that provides the basis for many of the sampling algorithms we consider in this chapter.

The statements of sampling algorithms are remarkably simple compared to the methods for exact inference discussed in previous chapters, and their accuracy can be made arbitrarily high by increasing the number of sampled situations. However, the design of appropriate sampling methods may not be trivial as we may need to focus the sampling process on a set of situations that are of particular interest.

Automated reasoning has been receiving much interest from a number of fields, including philosophy, cognitive science, and computer science. In this chapter, we consider the particular interest of computer science in automated reasoning over the last few decades, and then focus our attention on probabilistic reasoning using Bayesian networks, which is the main subject of this book.

Automated reasoning

The interest in automated reasoning within computer science dates back to the very early days of artificial intelligence (AI), when much work had been initiated for developing computer programs for solving problems that require a high degree of intelligence. Indeed, an influential proposal for building automated reasoning systems was extended by John McCarthy shortly after the term “artificial intelligence” was coined [McCarthy, 1959]. This proposal, sketched in Figure 1.1, calls for a system with two components: a knowledge base, which encodes what we know about the world, and a reasoner (inference engine), which acts on the knowledge base to answer queries of interest. For example, the knowledge base may encode what we know about the theory of sets in mathematics, and the reasoner may be used to prove various theorems about this domain.

McCarthy's proposal was actually more specific than what is suggested by Figure 1.1, as he called for expressing the knowledge base using statements in a suitable logic, and for using logical deduction in realizing the reasoning engine; see Figure 1.2. McCarthy's proposal can then be viewed as having two distinct and orthogonal elements.

We address in this chapter a number of problems that arise in real-world applications, showing how each can be solved by modeling and reasoning with Bayesian networks.

Introduction

We consider a number of real-world applications in this chapter drawn from the domains of diagnosis, reliability, genetics, channel coding, and commonsense reasoning. For each one of these applications, we state a specific reasoning problem that can be addressed by posing a formal query with respect to a corresponding Bayesian network. We discuss the process of constructing the required network and then identify the specific queries that need to be applied.

There are at least four general types of queries that can be posed with respect to a Bayesian network. Which type of query to use in a specific situation is not always trivial and some of the queries are guaranteed to be equivalent under certain conditions. We define these query types formally in Section 5.2 and then discuss them and their relationships in more detail when we go over the various applications in Section 5.3.

The construction of a Bayesian network involves three major steps. First, we must decide on the set of relevant variables and their possible values. Next, we must build the network structure by connecting the variables into a DAG. Finally, we must define the CPT for each network variable. The last step is the quantitative part of this construction process and can be the most involved in certain situations.