Determining full conditional independence by low-order conditioning

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 all 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 all 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 independences 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. call 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