Positive definite completion problems for directed acyclic graphs

TL;DR

This paper investigates positive definite completion problems for matrices related to directed acyclic graph models, providing algorithms, theoretical characterizations, and explicit formulas for completions in Bayesian network contexts.

Contribution

It introduces new procedures and theoretical results for completing partial matrices within DAG-based covariance and inverse-covariance spaces, extending undirected graph results.

Findings

Provided fast algorithms for matrix completion in DAG models

Characterized DAGs that can always be completed

Derived closed-form expressions for inverse and determinant

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 directed acyclic graph models (also known as Bayesian networks). 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 analog of the positive definite completion result for undirected graphs in the context of directed acyclic graphs, and thus proceed to characterize the class of DAGs 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.