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 "
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.