Positive Definite Completion Problems for Bayesian Networks

Abstract

A positive definite completion problem pertains to determining whether the unspecified positions of a partial (or incomplete) matrix can be completed in a desired subclass of positive definite matrices. In this paper we study an important and new class of positive definite completion problems where the desired subclasses are the spaces of covariance and inverse-covariance matrices of probabilistic models corresponding to Bayesian networks (also known as directed acyclic graph models). We provide fast procedures that determine whether a partial matrix can be completed in either of these spaces and thereafter proceed to construct the completed matrices. We prove an analogue of the positive definite completion result for undirected graphs in the context of directed acyclic graphs, and thus proceed to characterize the class of directed acyclic graphs which can always be completed. We also proceed to give closed form expressions for the inverse and the determinant of a completed matrix as a function of only the elements of the corresponding partial matrix.