Dear Colleagues,

We continue our Colloquium.

Marc Loizeaux,

University of Tennessee at Chattanooga

Tuesday, March 4, Metro 161, 2:00 pm.

Sampling Perfectly from a Bayesian Cluster Model

Abstract. A Bayesian cluster model for spatial point processes is
presented, in which observations are assumed to cluster around a finite
collection of underlying landmarks. The prior and likelihood are modeled
as locally stable point processes. We also assume that, for all data
sets, the Papangelou conditional intensities for the likelihood have a
common lower bound. Under these mild conditions the posterior is shown
to be locally stable on its support. 

We obtain "snapshots" of our posterior distribution using Markov chain
Monte Carlo techniques. Typically, an MCMC sampler obtains samples from
distributions that are "close to" the desired distribution. But due to
the local stability of our posterior we are able to obtain an exact draw
from the desired distribution via an algorithm of Kendall and Møller.
Thus we discuss the idea of  "perfect" sampling, in which a chain is run
from the distant past, and at time T=0 a sample from the desired
distribution is achieved.

An application to disease clustering, using leukemia location data from
upstate New York, is presented. We also present potential future
applications, some of which served to motivate the model presented.

Most of the ideas in this talk will be presented with a goal towards
intuitive understanding for students with some knowledge of statistics
and calculus.