A BACKWARD SELECTION PROCEDURE FOR APPROXIMATING A DISCRETE PROBABILITY DISTRIBUTION BY DECOMPOSABLE MODELS

Decomposable (probabilistic) models are log-linear models generated by acyclic hypergraphs, and a number of nice properties enjoyed by them are known. In many applications the following selection problem naturally arises: given a probability distribution p over a finite set V of n discrete variables and a positive integer k, find a decomposable model with tree-width k that best fits p. If H is the generating hypergraph of a decomposable model and p(H) is the estimate of p under the model, we can measure the closeness of p(H) to p by the information divergence D(p : p(H)), so that the problem above reads: given p and k, find an acyclic, connected hypergraph H of tree-width k such that D(p : pH) is minimum. It is well-known that this problem is NP-hard. However, for k = 1 it was solved by Chow and Liu in a very efficient way; thus, starting from an optimal Chow-Liu solution, a few forward-selection procedures have been proposed with the aim at finding a 'good' solution for an arbitrary k. We propose a backward-selection procedure which starts from the (trivial) optimal solution for k = n - 1, and we show that, in a study case taken from literature, our procedure succeeds in finding an optimal solution for every k.