## The Annals of Statistics

### On Monte Carlo methods for estimating ratios of normalizing constants

#### Abstract

Recently, estimating ratios of normalizing constants has played an important role in Bayesian computations. Applications of estimating ratios of normalizing constants arise in many aspects of Bayesian statistical inference. In this article, we present an overview and discuss the current Monte Carlo methods for estimating ratios of normalizing constants. Then we propose a new ratio importance sampling method and establish its theoretical framework. We find that the ratio importance sampling method can be better than the current methods, for example, the bridge sampling method (Meng and Wong) and the path sampling method (Gelman and Meng), in the sense of minimizing asymptotic relative mean-square errors of estimators. An example is given for illustrative purposes. Finally, we present two special applications and the general implementation issues for estimating ratios of normalizing constants.

#### Article information

Source
Ann. Statist., Volume 25, Number 4 (1997), 1563-1594.

Dates
First available in Project Euclid: 9 September 2002

https://projecteuclid.org/euclid.aos/1031594732

Digital Object Identifier
doi:10.1214/aos/1031594732

Mathematical Reviews number (MathSciNet)
MR1463565

Zentralblatt MATH identifier
0936.62028

Subjects
Primary: 62E25
Secondary: 62A15 62A99: None of the above, but in this section

#### Citation

Chen, Ming-Hui; Shao, Qi-Man. On Monte Carlo methods for estimating ratios of normalizing constants. Ann. Statist. 25 (1997), no. 4, 1563--1594. doi:10.1214/aos/1031594732. https://projecteuclid.org/euclid.aos/1031594732

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• WORCESTER, MASSACHUSETTS 01609-2280 EUGENE, OREGON 97403-1222 E-MAIL: mhchen@wpi.edu E-MAIL: qmshao@darkwing.uoregon.edu