PARTITION-WEIHTED MONTE CARLO ESTIMATION

MING-HUI CHEN^{1}, AND QI-MAN SHAO^{2}

1Department of Statistics, University of Connecticut, 215 Glenbrook Road, U-4120, Storrs,

CT 06269-4120, U.S.A., e-mail: mhchen@merlot.stat.uconn.edu

2Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, U.S.A., e-mail: qmshao@darkwing.uoregon.edu

(Received October 1, 1999; revised July 27, 2000)

Abstract. Although various efficient and sophisticated Markov chain Monte Carlo sampling methods have been developed during the last decade, the sample mean is still a dominant in computing Bayesian posterior quantities. The sample mean is simple, but may not be efficient. The weighted sample mean is a natural generalization of the sample mean. In this paper, a new weighted sample mean is proposed by partitioning the support of posterior distribution, so that the same weight is assigned to observations that belong to the same subset in the partition. A novel application of this new weighted sample mean in computing ratios of normalizing constants and necessary theory are provided. Illustrative examples are given to demonstrate the methodology.

Key words and phrases: Bayesian computation, importance sampling, Markov chain Monte Carlo, posterior distribution, simulation.