Independence properties of directed markov fields

TLDR

The study investigates directed Markov fields on finite graphs without assuming positivity of densities. The authors introduce a criterion for conditional independence, termed the directed global Markov property. They prove the directed local and global Markov properties are equivalent, and that their criterion—easy to use, sharper than previous ones and equivalent to Pearl’s—is optimal and cannot be further refined.

Abstract

Abstract We investigate directed Markov fields over finite graphs without positivity assumptions on the densities involved. A criterion for conditional independence of two groups of variables given a third is given and named as the directed, global Markov property. We give a simple proof of the fact that the directed, local Markov property and directed, global Markov property are equivalent and – in the case of absolute continuity w. r. t. a product measure – equivalent to the recursive factorization of densities. It is argued that our criterion is easy to use, it is sharper than that given by Kiiveri, Speed, and Carlin and equivalent to that of Pearl. It follows that our criterion cannot be sharpened.