Independence properties of directed markov fields

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.

The BUGS project: Evolution, critique and future directions

“…The joint distribution of all nodes V in any DAG can be factorized as follows [29]: where P v denotes the set of (stochastic) parents of node v. With regard to Bayesian inference, the data D and the unknowns , say, together form a partition of V , and so the joint posterior density p( |D) ∝ p(V ). For the purposes of Gibbs sampling we are interested in the full conditional distributions of each unknown stochastic node conditional on the values of all other nodes in the graph.…”

Combining MCMC with ‘sequential’ PKPD modelling

“…A DAG expresses the assumption that any variable is conditionally independent of all its 'non-descendants', given its 'parents'. If we wish to define a joint distribution over all variables, V, say, in a given graph, such independence properties are equivalent [23] to assuming…”

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“…In the case of undirected graphs this criterion corresponds to check whether all paths between vertices in A and B intersect S. In such case A ⊥ ⊥ B | S [g] for an undirected graph g. For directed graphs the notion is less intuitive and we recommend the interested reader to consult Lauritzen et al (1990) or Pearl and Verma (1987).…”