Strong Completeness and Faithfulness in Bayesian Networks

TL;DR

This paper proves a strong measure-theoretic completeness result for d-separation in discrete Bayesian networks, showing that almost all distributions compatible with a given structure are faithful, meaning all and only the independencies implied by the structure hold.

Contribution

It establishes a strong measure-theoretic faithfulness result for discrete Bayesian networks, linking structure and distribution independence properties.

Findings

Almost all distributions are faithful to the network structure.

D-separation is a complete criterion for independence in discrete Bayesian networks.

Faithfulness holds in a strong measure-theoretic sense.

Abstract

A completeness result for d-separation applied to discrete Bayesian networks is presented and it is shown that in a strong measure-theoretic sense almost all discrete distributions for a given network structure are faithful; i.e. the independence facts true of the distribution are all and only those entailed by the network structure.