Skip to main content
×
Home
    • Aa
    • Aa

Ergodicity of Markov chain Monte Carlo with reversible proposal

  • K. Kamatani (a1)
Abstract
Abstract

We describe the ergodic properties of some Metropolis–Hastings algorithms for heavy-tailed target distributions. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned Crank–Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that, under a suitable transformation, the MpCN algorithm becomes a random-walk Metropolis algorithm.

Copyright
Corresponding author
* Postal address: Graduate School of Engineering Science and Center for Mathematical Modeling and Data Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan. Email address: kamatani@sigmath.es.osaka-u.ac.jp
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] Y. Atchadé and G. Fort (2010). Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. Bernoulli 16, 116154.

[2] A. Beskos , G. Roberts , A. Stuart and J. Voss (2008). MCMC methods for diffusion bridges. Stoch. Dyn. 8, 319350.

[6] P. Dellaportas and G. O. Roberts (2003). An introduction to MCMC. In Spatial Statistics and Computational Methods (Aalborg, 2001; Lecture Notes Statist. 173), Springer, New York, pp. 141.

[8] S. Dutta (2012). Multiplicative random walk Metropolis-Hastings on the real line. Sankhyā B 74, 315342.

[9] G. Fort and E. Moulines (2000). V-subgeometric ergodicity for a Hastings-Metropolis algorithm. Statist. Prob. Lett. 49, 401410.

[10] G. Fort and E. Moulines (2003). Polynomial ergodicity of Markov transition kernels. Stoch. Process. Appl. 103, 5799.

[11] M. Hairer , A. M. Stuart and S. J. Vollmer (2014). Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions. Ann. Appl. Prob. 24, 24552490.

[12] Z. Hu , Z. Yao and J. Li (2017). On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inference. J. Comput. Phys. 332, 492503.

[16] S. F. Jarner and R. L. Tweedie (2003). Necessary conditions for geometric and polynomial ergodicity of random-walk-type Markov chains. Bernoulli 9, 559578.

[17] L. T. Johnson and C. J. Geyer (2012). Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm. Ann. Statist. 40, 30503076.

[18] K. Kamatani (2009). Metropolis-Hastings algorithms with acceptance ratios of nearly 1. Ann. Inst. Statist. Math. 61, 949967.

[20] K. Kamatani and M. Uchida (2015). Hybrid multi-step estimators for stochastic differential equations based on sampled data. Statist. Inf. Stoch. Process. 18, 177204.

[26] E. Nummelin (1984). General Irreducible Markov Chains and Nonnegative Operators (Camb. Tracts Math. 83). Cambridge University Press.

[28] G. O. Roberts and R. L. Tweedie (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83, 95110.

[30] L. Tierney (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22, 17011762.

[31] P. Tuominen and R. L. Tweedie (1994). Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Prob. 26, 775798.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 17 *
Loading metrics...

Abstract views

Total abstract views: 78 *
Loading metrics...

* Views captured on Cambridge Core between 22nd June 2017 - 23rd September 2017. This data will be updated every 24 hours.