Causal models have no complete axiomatic characterization

TL;DR

This paper demonstrates that causal models represented by Bayesian networks lack a complete axiomatic characterization, unlike Markov networks, due to their non-closure under sub-models.

Contribution

It proves that Bayesian network independencies cannot be fully axiomatized using Horn or disjunctive clauses, highlighting fundamental differences from Markov networks.

Findings

Markov networks have a finite axiomatic characterization.

Bayesian networks lack a complete axiomatization.

Sub-models of causal models may not be causal.

Abstract

Markov networks and Bayesian networks are effective graphic representations of the dependencies embedded in probabilistic models. It is well known that independencies captured by Markov networks (called graph-isomorphs) have a finite axiomatic characterization. This paper, however, shows that independencies captured by Bayesian networks (called causal models) have no axiomatization by using even countably many Horn or disjunctive clauses. This is because a sub-independency model of a causal model may be not causal, while graph-isomorphs are closed under sub-models.