The Annals of Applied Probability

A note on Metropolis-Hastings kernels for general state spaces

Luke Tierney

Full-text: Open access


The Metropolis-Hastings algorithm is a method of constructing a reversible Markov transition kernel with a specified invariant distribution. This note describes necessary and sufficient conditions on the candidate generation kernel and the acceptance probability function for the resulting transition kernel and invariant distribution to satisfy the detailed balance conditions. A simple general formulation is used that covers a range of special cases treated separately in the literature. In addition, results on a useful partial ordering of finite state space reversible transition kernels are extended to general state spaces and used to compare the performance of two approaches to using mixtures in Metropolis-Hastings kernels.

Article information

Ann. Appl. Probab. Volume 8, Number 1 (1998), 1-9.

First available in Project Euclid: 29 July 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 65C05: Monte Carlo methods 62-04: Explicit machine computation and programs (not the theory of computation or programming)

Markov chain Monte Carlo Peskun's theorem mixture kernels


Tierney, Luke. A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab. 8 (1998), no. 1, 1--9. doi:10.1214/aoap/1027961031.

Export citation


  • Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation (with discussion). J. Roy. Statist. Soc. Ser. B 55 25-37.
  • Besag, J., Green, P. J., Higdon, D. and Mengersen, K. (1995). Bayesian computation and stochastic sy stems (with discussion). Statist. Sci. 10 3-66.
  • Green, P. J. (1995). Reversible jump MCMC computation and Bayesian model determination. Biometrika 82 711-732.
  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phy s. 104 1-19. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E.
  • (1953). Equations of state calculations by fast computing machines. J. Chemical physics 21 1087-1091.
  • Peskun, P. H. (1973). Optimum Monte Carlo sampling using Markov chains. Biometrika 60 607- 612.
  • Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. Ser. B 55 3-24.
  • Tierney, L. (1991). Markov chains for exploring posterior distributions. Technical Report 560, School of Statistics, Univ. Minnesota.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701- 1786.