Using Tree-Decomposable Structures to Approximate Belief Networks

TL;DR

This paper introduces a method for approximating belief networks using tree-decomposable structures, enhancing computational efficiency and accuracy in belief propagation.

Contribution

It proposes the concept of tree-decomposable structures supported by star decomposition, and develops greedy and exact algorithms for optimal approximation.

Findings

Tree-decomposable structures support a wider class of distributions.

The logarithm scoring rule effectively evaluates approximation quality.

Algorithms for finding optimal approximations are developed and analyzed.

Abstract

Tree structures have been shown to provide an efficient framework for propagating beliefs [Pearl,1986]. This paper studies the problem of finding an optimal approximating tree. The star decomposition scheme for sets of three binary variables [Lazarsfeld,1966; Pearl,1986] is shown to enhance the class of probability distributions that can support tree structures; such structures are called tree-decomposable structures. The logarithm scoring rule is found to be an appropriate optimality criterion to evaluate different tree-decomposable structures. Characteristics of such structures closest to the actual belief network are identified using the logarithm rule, and greedy and exact techniques are developed to find the optimal approximation.