D-Separation and computation of probability distributions in Bayesian networks

摘要

Consider a family \({(X_i)_{i \in I}}\) of random variables endowed with the structure of a Bayesian network, and a subset S of I. This paper examines the problem of computing the probability distribution of the subfamily \({(X_{a})_{a \in S}}\) (respectively the probability distribution of \({ (X_{b})_{b \in {\bar{S}}}}\) , where \({{\bar{S}} = I - S}\) , conditional on \({(X_{a})_{a \in S}}\)). This paper presents some theoretical results that makes it possible to compute joint and conditional probabilities over a subset of variables by computing over separate components. In other words, it is demonstrated that it is possible to decompose this task into several parallel computations, each related to a subset of S (respectively of \({{\bar{S}}}\)); these partial results are then put together as a final product. In computing the probability distribution over \({(X_a)_{a \in S}}\) , this procedure results in the production of a structure of level two Bayesian network structure for S.