D-Separation and computation of probability distributions in Bayesian networks

摘要

Consider a family (X_i )_(i∈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∈S) (respectively the probability distribution of (X_b)_(b∈S{top}-), where S{top}- = I - S, conditional on (X_a)_(a∈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 5); these partial results are then put together as a final product. In computing the probability distribution over (X_a)_(a∈s), this procedure results in the production of a structure of level two Bayesian network structure for S.