Building Hyper Dirichlet Processes for Graphical Models

TL;DR

This paper introduces a novel class of hyper Dirichlet processes tailored for graphical models, enabling flexible Bayesian modeling of complex conditional independence structures in multivariate data.

Contribution

It develops a new non-parametric prior based on hyper Dirichlet processes for graphical models, extending existing Bayesian frameworks for more flexible dependence modeling.

Findings

Provides a new non-parametric prior for graphical models

Enables Bayesian inference with complex dependence structures

Extends Dirichlet process methods to graphical model settings

Abstract

Graphical models are used to describe the conditional independence relations in multivariate data. They have been used for a variety of problems, including log-linear models (Liu and Massam, 2006), network analysis (Holland and Leinhardt, 1981; Strauss and Ikeda, 1990; Wasserman and Pattison, 1996; Pattison and Wasserman, 1999; Robins et al., 1999);, graphical Gaussian models (Roverato and Whittaker, 1998; Giudici and Green, 1999; Marrelec and Benali, 2006), and genetics (Dobra et al., 2004). A distribution that satisfies the conditional independence structure of a graph is Markov. A graphical model is a family of distributions that is restricted to be Markov with respect to a certain graph. In a Bayesian problem, one may specify a prior over the graphical model. Such a prior is called a hyper Markov law if the random marginals also satisfy the independence constraints. Previous work in this area includes (Dempster, 1972; Dawid and Lauritzen, 1993; Giudici and Green, 1999; Letac and Massam, 2007). We explore graphical models based on a non-parametric family of distributions, developed from Dirichlet processes.