Hyper Markov law in undirected graphical models with its applications

By exploring the prime decomposition of undirected graphs, this work investigates the hyper Markov property within the framework of arbitrary undirected graph, which can be seen as the generalization of that for decomposable graphical models proposed by Dawid and Laurizten. The proposed hyper Markov properties of this article can be used to characterize the conditional independence of a distribution or a statistical quantity and helpful to simplify the likelihood ratio functions for statistical test for two different graphs obtained by removing or adding one edge. As an application of these properties, the G-Wishart law is introduced as a prior law for graphical Gaussian models for Bayesian posterior updating, and a hypothesis test for precision matrix is designed to determine the model structures. Our simulation experiments are implemented by using the Gibbs Sampler algorithm and the results show that it performs better in convergence speed.