Convergence rates and asymptotic standard errors for MCMC
algorithms for Bayesian probit regression 
Vivekananda Roy , Ph.D. Candidate
University of Florida
3:00 p.m. February 1, 2008
ABSTRACT
We study Markov chain Monte Carlo algorithms for exploring
the intractable posterior density that results when a probit
regression likelihood is combined with a flat prior on the
regression coefficient. We prove that the data augmentation
algorithm of Albert and Chib (1993) and the PXDA algorithm of Liu
and Wu (1999) both converge at a geometric rate, which ensures the
existence of central limit theorems (CLTs) for ergodic averages
under a second moment condition. While these two algorithms are
essentially equivalent in terms of computational complexity, we show
that the PXDA algorithm is theoretically more efficient in the
sense that the asymptotic variance in the CLT under the PXDA
algorithm is no larger than that under Albert and Chib's algorithm.
A simple, consistent estimator of the asymptotic variance in the CLT
is constructed using regeneration. As an illustration, we apply our
results to the lupus data from van Dyk and Meng (2001). In this
particular example, the estimated asymptotic relative efficiency of
the PXDA algorithm with respect to Albert and Chib's algorithm is
about 65, which demonstrates that huge gains in efficiency are
possible by using PXDA algorithm.
