Alternative parametrizations and reference priors for decomposable discrete graphical models

TL;DR

This paper explores alternative parameterizations and constructs reference priors for decomposable discrete graphical models, enhancing Bayesian analysis by providing order-independent, proper, and conjugate priors for various parameter groupings.

Contribution

It introduces new parametrizations based on clique-residuals and log-odds-ratios, along with order-independent reference priors for these models.

Findings

Reference priors are proper and conjugate.

Priors do not depend on parameter grouping order.

New parametrizations improve Bayesian inference.

Abstract

For a given discrete decomposable graphical model, we identify several alternative parametrizations, and construct the corresponding reference priors for suitable groupings of the parameters. Specifically, assuming that the cliques of the graph are arranged in a perfect order, the parameters we consider are conditional probabilities of clique-residuals given separators, as well as generalized log-odds-ratios. We also consider a parametrization associated to a collection of variables representing a cut for the statistical model. The reference priors we obtain do not depend on the order of the groupings, belong to a conjugate family, and are proper.