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Structural Markov graph laws for Bayesian model uncertainty. (English) Zbl 1317.62046

Summary: This paper considers the problem of defining distributions over graphical structures. We propose an extension of the hyper Markov properties of A. P. Dawid and S. L. Lauritzen [Ann. Stat. 21, No. 3, 1272–1317 (1993; Zbl 0815.62038)], which we term structural Markov properties, for both undirected decomposable and directed acyclic graphs, which requires that the structure of distinct components of the graph be conditionally independent given the existence of a separating component. This allows the analysis and comparison of multiple graphical structures, while being able to take advantage of the common conditional independence constraints. Moreover, we show that these properties characterise exponential families, which form conjugate priors under sampling from compatible Markov distributions.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
05C80 Random graphs (graph-theoretic aspects)
05C90 Applications of graph theory
68T30 Knowledge representation

Citations:

Zbl 0815.62038

Software:

HdBCS; Separoids
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References:

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