## A note on Metropolis-Hastings kernels for general state spaces.(English)Zbl 0935.60053

Let $$P(x,dy)$$ be a Markov transition kernel on a measurable space $$E$$ with a specified invariant distribution $$\pi$$. Consider the Hastings-Metropolis kernel $P(x,dy)= Q(x,dy)\alpha (x,y)+ \delta_x(dy) \int\bigl(1-\alpha (x,u)\bigr) Q(x,du),$ where $$Q(x,dy)$$ is a transition kernel, $$\alpha (x,y): E\times E\to[0,1]$$, and $$\delta_x$$ is point mass at $$x$$. First, necessary and sufficient conditions on $$Q$$ and $$\alpha$$ are given for the reversibility of this type of kernel. Next, in order to compare the performances of such kernels, a result of P. H. Peskun [Biometrika 60, 607-612 (1973; Zbl 0271.62041)] on the ordering of asymptotic variances is extended from finite to general spaces.

### MSC:

 60J05 Discrete-time Markov processes on general state spaces 65C05 Monte Carlo methods

Zbl 0271.62041
Full Text:

### References:

 [1] Besag, J. and Green, P. J. (1993). Spatial statistics and Bayesian computation (with discussion). J. Roy. Statist. Soc. Ser. B 55 25-37. JSTOR: · Zbl 0800.62572 [2] Besag, J., Green, P. J., Higdon, D. and Mengersen, K. (1995). Bayesian computation and stochastic sy stems (with discussion). Statist. Sci. 10 3-66. · Zbl 0955.62552 [3] Green, P. J. (1995). Reversible jump MCMC computation and Bayesian model determination. Biometrika 82 711-732. JSTOR: · Zbl 0861.62023 [4] Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109. · Zbl 0219.65008 [5] 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. · Zbl 0588.60058 [6] . Equations of state calculations by fast computing machines. J. Chemical physics 21 1087-1091. · Zbl 0222.73111 [7] Peskun, P. H. (1973). Optimum Monte Carlo sampling using Markov chains. Biometrika 60 607- 612. JSTOR: · Zbl 0271.62041 [8] 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. JSTOR: · Zbl 0779.62030 [9] Tierney, L. (1991). Markov chains for exploring posterior distributions. Technical Report 560, School of Statistics, Univ. Minnesota. · Zbl 0829.62080 [10] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701- 1786. · Zbl 0829.62080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.