The Annals of Statistics

The pseudo-marginal approach for efficient Monte Carlo computations

Christophe Andrieu and Gareth O. Roberts

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We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139–1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.

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Ann. Statist. Volume 37, Number 2 (2009), 697-725.

First available in Project Euclid: 10 March 2009

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Zentralblatt MATH identifier

Primary: 60J22: Computational methods in Markov chains [See also 65C40] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Markov chain Monte Carlo auxiliary variable marginal convergence


Andrieu, Christophe; Roberts, Gareth O. The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Statist. 37 (2009), no. 2, 697--725. doi:10.1214/07-AOS574.

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  • [1] Beaumont, M. A. (2003). Estimation of population growth or decline in genetically monitored populations. Genetics 164 1139–1160.
  • [2] Dellaportas, P. and Forster, J. J. (1999). Markov chain Monte Carlo model determination for hierarchical and graphical log-linear models. Biometrika 86 615–633.
  • [3] Green, P. J. (1995). Reversible jump MCMC computation and Bayesian model determination. Biometrika 82 711–732.
  • [4] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.
  • [5] Liu, J. S., Wong, W. H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika 81 27–40.
  • [6] Ntzoufras, I., Dellaportas, P. and Forster, J. (2003). Bayesian variable and link determination for generalized linear models. J. Statist. Plann. Inference 111 165–180.
  • [7] O’Neill, P. D., Balding, D. J., Becker, N. G., Eerola, M. and Mollison, D. (2000). Analyzes of infectious disease data from houseing the expected value of ratios, hold outbreaks by Markov chain Monte Carlo methods. Appl. Statist. 49 517–542.
  • [8] Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • [9] Roberts, G. O. and Sahu, S. K. (1997). Updating schemes, correlation structure, blocking and parameterisation for the Gibbs sampler. J. Roy. Static. Soc. Ser. B 59 291–397.
  • [10] Roberts, G. O. and Tweedie, R. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95–110.
  • [11] Tierney, L. (1998). A note on Metropolis–Hastings kernels for general state-spaces. Ann. Appl. Probab. 8 1–9.