Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models

TLDR

The study revisits Markov probability properties with an abstract approach applicable to hyper Markov laws, noting their natural occurrence in decomposable log‑linear and covariance selection models. The paper introduces and investigates hyper Markov laws, probability distributions over probability measures concentrated on Markov probabilities of decomposable graphs and satisfying related conditional independence restrictions. The authors define hyper Markov laws, study a stronger property, and construct specific examples such as hyper multinomial, hyper Dirichlet, hyper Wishart, and inverse Wishart, using an abstract approach to Markov probabilities. It is shown constructively that hyper Markov laws exist, serve as sampling distributions of maximum likelihood estimators in decomposable graphical models, and act as natural conjugate priors for Bayesian analysis of these models.

Abstract

This paper introduces and investigates the notion of a hyper Markov law, which is a probability distribution over the set of probability measures on a multivariate space that (i) is concentrated on the set of Markov probabilities over some decomposable graph, and (ii) satisfies certain conditional independence restrictions related to that graph. A stronger version of this hyper Markov property is also studied. Our analysis starts by reconsidering the properties of Markov probabilities, using an abstract approach which thereafter proves equally applicable to the hyper Markov case. Next, it is shown constructively that hyper Markov laws exist, that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models, and also that they form natural conjugate prior distributions for a Bayesian analysis of these models. As examples we construct a range of specific hyper Markov laws, including the hyper multinomial, hyper Dirichlet and the hyper Wishart and inverse Wishart laws. These laws occur naturally in connection with the analysis of decomposable log-linear and covariance selection models.