The Annals of Statistics

On Monte Carlo methods for estimating ratios of normalizing constants

Ming-Hui Chen and Qi-Man Shao

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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

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

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Bayesian computation bridge sampling Gibbs sampling importance sampling Markov chain Monte Carlo Metropolis-Hastings algorithm path sampling ratio importance sampling


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.

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