Separation Properties of Sets of Probability Measures
This work addresses foundational issues in probabilistic graphical models for researchers in AI and statistics, though it appears incremental as it builds on existing concepts like d-separation and epistemic independence.
The paper tackles the problem of extending separation properties from Bayesian networks to sets of probability measures with epistemic independence, showing that a proposed strong Markov condition enforces strong independence and separation properties, implying these properties extend to epistemic independence.
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.