A Pair of Bayesian Network Structures has Undecidable Conditional Independencies

TL;DR

This paper proves that determining whether two Bayesian network structures imply a specific conditional independence is undecidable, highlighting fundamental limitations in combining Bayesian network knowledge.

Contribution

It establishes the undecidability of conditional independence inference from pairs of Bayesian network structures, a problem previously assumed to be tractable.

Findings

Undecidability of conditional independence inference for two Bayesian networks

Explicit construction of network pairs with unprovable implications in ZFC

Fundamental limits on combining Bayesian network structures

Abstract

Given a Bayesian network structure (directed acyclic graph), the celebrated d-separation algorithm efficiently determines whether the network structure implies a given conditional independence relation. We show that this changes drastically when we consider two Bayesian network structures instead. It is undecidable to determine whether two given network structures imply a given conditional independency, that is, whether every collection of random variables satisfying both network structures must also satisfy the conditional independency. Although the approximate combination of two Bayesian networks is a well-studied topic, our result shows that it is fundamentally impossible to accurately combine the knowledge of two Bayesian network structures, in the sense that no algorithm can tell what conditional independencies are implied by the two network structures. We can also explicitly construct two Bayesian network structures, such that whether they imply a certain conditional independency is unprovable in the ZFC set theory, assuming ZFC is consistent.