Dr. Gerard Letac

Department of Mathematics

University of Paul Sabatier, Toulouse, France

* Title:* " Polytope of a hierarchical model for a contingency table and applications to Bayes factors "

* Abstract:*

A contingency table with k criterias is a multinomial random variable indexed

by the product I of k finite sets and defined by an unknown probability p on I,

to be estimated. Given a family D of subsets of the k criterias such that two

elements of D are never included in each other, the hierarchical model governed

by D is the set of probabilities p on I such that log p (i) is a sum of functions

f(d)(i) where d runs on D and where f(d)(i) depends only on components of i which

are in d. This model is actually a general exponential family. We choose to put

on this model an a priori probability which is in the Diaconis Ylvisaker family

of the associated natural exponential family (NEF). This DY family depends on a

parameter m located in the domain of the means M of the NEF (M is the polytope of

the title) and on a positive parameter a. The normalizing constant of this a priori

distribution is the inverse of an integral I(m,a). When a tends to zero we prove

that I(m,a) is equivalent to a power of a, multiplied by the volume of the polar

set of M-m. We use this result and various refinements of it to compare two

hierarchical models on I generated by two different families D and D': the Bayes

factor associated to this model selection problem is actually a ratio between

products of functions of the form I(m,a) and the above results enable us to study

the limit of this Bayes factor when a tends to zero.

This is a work done with Professor Helene Massam.