Bayes Factors and the Geometry of Discrete Loglinear Models

• Activity Number: 373
• Type: Invited
• Date/Time: Tuesday, July 31, 2012 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Helene M Massam*+ and Gerard Letac
• Companies: York University and Universite Paul Sabatier
• Keywords: discrete loglinear models ; Bayes factors ; geometry ; characteristic function ; faces of a polyhedron

Abstract

We consider the class of hierarchical loglinear models for discrete data with multinomial sampling. We assume that the Diaconis-Ylvisaker conjugate prior is the prior distribution on the loglinear parameters. Under these conditions, the Bayes factor between two models is a function of their prior and posterior normalizing constants under each model. These constants are functions of the data and of the hyperparameters $(m,\alpha)$, which can be interpreted respectively as marginal counts and the total count of a fictive contingency table.

Both constants are finite when $\alpha$ is positive and $m$ and the data are in the interior $C$ of the support of the multinomial distribution. We will always assume that $m$ which is a prior hyperparameter of our choice satisfies this condition and we study what happens when $\alpha\rightarrow 0$ and the data is on the boundary of $C$. We will see that the behaviour of the Bayes factor is dictated by the dimension of the face of $C$ to which the data belongs. We will also emphasize the role played by the characteristic function of $C$, a new object in the toolbox of exponential families.