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November 2009 Determining full conditional independence by low-order conditioning
Dhafer Malouche
Bernoulli 15(4): 1179-1189 (November 2009). DOI: 10.3150/09-BEJ193


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.


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Dhafer Malouche. "Determining full conditional independence by low-order conditioning." Bernoulli 15 (4) 1179 - 1189, November 2009.


Published: November 2009
First available in Project Euclid: 8 January 2010

zbMATH: 1206.60009
MathSciNet: MR2597588
Digital Object Identifier: 10.3150/09-BEJ193

Rights: Copyright © 2009 Bernoulli Society for Mathematical Statistics and Probability


Vol.15 • No. 4 • November 2009
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