Maximum likelihood bounded tree-width Markov networks

It is proved that the problem of projecting a distribution onto (or finding a maximum likelihood distribution among) Markov networks of bounded tree-width is NP-hard to solve exactly and an approximation algorithm with a provable performance guarantee is provided.

Abstract

Chow and Liu (1968) studied the problem of learning a maximumlikelihood Markov tree. We generalize their work to more complexMarkov networks by considering the problem of learning a maximumlikelihood Markov network of bounded complexity. We discuss howtree-width is in many ways the appropriate measure of complexity andthus analyze the problem of learning a maximum likelihood Markovnetwork of bounded tree-width.Similar to the work of Chow and Liu, we are able to formalize thelearning problem as a combinatorial optimization problem on graphs. Weshow that learning a maximum likelihood Markov network of boundedtree-width is equivalent to finding a maximum weight hypertree. Thisequivalence gives rise to global, integer-programming based,approximation algorithms with provable performance guarantees, for thelearning problem. This contrasts with heuristic local-searchalgorithms which were previously suggested (e.g. by Malvestuto 1991).The equivalence also allows us to study the computational hardness ofthe learning problem. We show that learning a maximum likelihoodMarkov network of bounded tree-width is NP-hard, and discuss thehardness of approximation.