Local Computation

Abstract

In this chapter we discuss different ways to compute marginals of a valuation which factors into a combination of valuations. We have seen in Chapter 2 that the complexity of the operations of combination and marginalization tends to increase exponentially with the size of the domains of the valuations involved. A crude measure of the size of a domain s is its cardinality |s|, the number of variables in the domain. A better measure would be the cardinality of the frame |Ω^s|. So we are interested in methods where the operations needed to compute a marginal can be limited to small domains. This basic problem has been recognized early in the development of Bayesian networks. (Lauritzen & Spiegelhalter, 1988) was the pioneering work which showed how join trees (also called junction trees, or Markov trees) can be used to compute marginals of large multidimensional discrete probability distributions, if they factor into factors with small domains. Shenoy and Shafer (Shenoy & Shafer 1990) introduced the axioms of valuation algebras needed to generalize computation on join trees from probability to other formalisms, especially belief functions. In the sequel many refinements and several different architectures for computing on join trees have been proposed. We refer to (Shafer, 1996) and (Cowell et. al., 1999; Dawid, 1992) for a discussion of this subject relative to probability theory. Relatively few contributions appeared concerning computation on join trees for other formalisms than probability theory. One notable exception is the paper (Lauritzen & Jensen, 1997) which discusses the most important architectures in the context of abstract valuation algebras. We refer also to (Dechter, 1999) for related algorithms, which can be applied to a number of different formalisms.