Separation Properties of Sets of Probability Measures

TL;DR

This paper investigates the separation properties of sets of probability measures in Bayesian networks, proposing a strong Markov condition that ensures separation and strong independence, thus extending classical properties to epistemic independence.

Contribution

It introduces a strong Markov condition that guarantees separation properties and strong independence for sets of probability measures, addressing limitations of existing independence concepts.

Findings

Strong Markov condition enforces separation properties.

Epistemic independence combined with the strong Markov condition leads to strong independence.

Separation properties of Bayesian networks extend to epistemic independence under the strong Markov condition.

Abstract

This paper analyzes independence concepts for sets of probability measures associated with directed acyclic graphs. The paper shows that epistemic independence and the standard Markov condition violate desirable separation properties. The adoption of a contraction condition leads to d-separation but still fails to guarantee a belief separation property. To overcome this unsatisfactory situation, a strong Markov condition is proposed, based on epistemic independence. The main result is that the strong Markov condition leads to strong independence and does enforce separation properties; this result implies that (1) separation properties of Bayesian networks do extend to epistemic independence and sets of probability measures, and (2) strong independence has a clear justification based on epistemic independence and the strong Markov condition.