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 PX-DA 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 PX-DA algorithm is theoretically more efficient in the
 sense that the asymptotic variance in the CLT under the PX-DA
 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 PX-DA algorithm with respect to Albert and Chib's algorithm is
 about 65, which demonstrates that huge gains in efficiency are
 possible by using PX-DA algorithm.