Determining full conditional independence by low-order conditioning

TL;DR

This paper proves that for perfect Markovian distributions, the full conditional independence graph can be identified by conditioning on a limited number of variables, specifically the size of the largest minimal separator.

Contribution

It establishes that the concentration graph of a perfect Markov distribution can be determined by low-order conditioning, equal to the maximum size of minimal separators.

Findings

Full conditional independence graph can be identified with limited conditioning.

Number of variables to condition on equals the largest minimal separator size.

Applicable to perfect Markovian distributions.

Abstract

A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In the multivariate Gaussian case, the absence of an edge corresponds to a zero coefficient in the precision matrix, which is the inverse of the covariance matrix. It is well known that this concentration graph represents some of the conditional independencies in the distribution of the associated random vector. These conditional independencies correspond to the "separations" or absence of edges in that graph. In this paper we assume that there are no other independencies present in the probability distribution than those represented by the graph. This property is called the perfect Markovianity of the probability distribution with respect to the associated concentration graph. We prove in this paper that this particular concentration graph, the one associated with a perfect Markov distribution, can be determined by only conditioning on a limited number of variables. We demonstrate that this number is equal to the maximum size of the minimal separators in the concentration graph.