Cornell University

Friday, March 7,1997

3:30 pm - refreshments

4:00 pm - talk

Room 327, Yost Hall

Abstract

The relationship between theory and computation has always been part of the business of statisticians, as the thorough solution to a problem usually requires a bit of both. This relationship is examined in some detail, with particular attention to computation as performed using Markov chain Monte Carlo algorithms. We see that there is not only interesting interplay between the mathematics of the statistical theory and the numerical algorithms, but also such interplay leads naturally to considering both Bayesian and frequentist approaches to a problem.

This work was originally motivated by a problem encountered by a practitioner, whose Gibbs sampling program suddenly was giving "crazy" answers, after giving nice answers for over a year. This problem was the start of a number of investigations into the statistical properties of the output of computational algorithms, and led to questions such as

the effect, on the Gibbs sampler, of improper prior and posterior distributions,

+ how to recognize such situations (difficult in practice!) and what to do with them (be

careful!),

the effect of Rao-Blackwellization, not only in the now-familiar case of Gibbs sampling but

also in the case of Metropolis-like algorithms. ~here the Ra~Blackwellization has a

nonparametric flavor.)

balancing statistical and computational improvements (full Rao-Blackwellization can bring

along an enormous computational burden).

Incidentally, the practitioner, whose problem started this entire investigation, had a model that resulted in a null Markov chain, and his Gibbs sampler (for over a year) produced lovely pictures of nonexistent posterior distributions.