Skip to main content
SpringerLink
Log in

Ergodicity of Combocontinuous Adaptive MCMC Algorithms

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

This paper proves convergence to stationarity of certain adaptive MCMC algorithms, under certain assumptions including easily-verifiable upper and lower bounds on the transition densities and a continuous target density. In particular, the transition and proposal densities are not required to be continuous, thus improving on the previous ergodicity results of Craiu et al. (Ann Appl Probab 25(6):3592–3623, 2015).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Andrieu C, Atchadé YF (2007) On the efficiency of adaptive MCMC algorithms. Electron Commun Probab 12(33):336–349

    Article  MathSciNet  MATH  Google Scholar 

  • Andrieu C, Moulines E (2006) On the ergodicity properties of some adaptive Markov chain Monte Carlo algorithms. Ann Appl Probab 16(3):1462–1505

    Article  MathSciNet  MATH  Google Scholar 

  • Andrieu C, Thoms J (2008) A tutorial on adaptive MCMC. Stat Comput 18:343–373

    Article  MathSciNet  Google Scholar 

  • Atchadé YF, Rosenthal JS (2005) On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11(5):815–828

    Article  MathSciNet  MATH  Google Scholar 

  • Bai Y, Roberts GO, Rosenthal JS (2011) On the containment condition for adaptive Markov chain Monte Carlo algorithms. Adv Appl Stat 21(1):1–54

    MathSciNet  MATH  Google Scholar 

  • Brooks S, Gelman A, Jones GL, Meng X (eds) (2011) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Craiu RV, Gray L, Latuszynski K, Madras N, Roberts GO, Rosenthal JS (2015) Stability of adversarial markov chains, with an application to adaptive mcmc algorithms. Ann Appl Probab 25(6):3592–3623

    Article  MathSciNet  MATH  Google Scholar 

  • Fort G, Moulines E, Priouret P (2011) Convergence of adaptive and interacting Markov chain Monte Carlo algorithms. Ann Stat 39(6):3262–3289

    Article  MathSciNet  MATH  Google Scholar 

  • Gaver DP, O’Muircheartaigh IG (1987) Robust empirical Bayes analyses of event rates. Technometrics 29(1):1–15

    Article  MathSciNet  MATH  Google Scholar 

  • George EI, Makov UE, Smith AFM (1993) Conjugate likelihood distributions. Scand J Stat 20(2):147–156

    MathSciNet  MATH  Google Scholar 

  • Giordani P, Kohn R (2010) Adaptive independent Metropolis–Hastings by fast estimation of mixtures of normals. J Comput Graph Sta. 19(2):243–259

    Article  MathSciNet  Google Scholar 

  • Haario H, Saksman E, Tamminen J (2001) An adaptive Metropolis algorithm. Bernoulli 7(2):223–242

    Article  MathSciNet  MATH  Google Scholar 

  • Haario H, Laine M, Mira A, Saksman E (2006) DRAM: efficient adaptive MCMC. Stat Comput 16(4):339–354

    Article  MathSciNet  Google Scholar 

  • Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109

    Article  MathSciNet  MATH  Google Scholar 

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092

    Article  Google Scholar 

  • Roberts GO, Rosenthal JS (2004) General state space Markov chains and MCMC algorithms. Probab Surv 1:20–71

    Article  MathSciNet  MATH  Google Scholar 

  • Roberts GO, Rosenthal JS (2006) Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains. Ann Appl Probab 16(4):2123–2139

    Article  MathSciNet  MATH  Google Scholar 

  • Roberts GO, Rosenthal JS (2007) Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J Appl Probab 44(2):458–475

    Article  MathSciNet  MATH  Google Scholar 

  • Roberts GO, Rosenthal JS (2009) Examples of adaptive MCMC. J Comput Graph Stat 18(2):349–367

    Article  MathSciNet  Google Scholar 

  • Rosenthal JS (2004) Adaptive MCMC Java applet. http://probability.ca/jeff/java/adapt.html

  • Rudin W (1976) Principles of mathematical analysis, 3rd edn. McGraw-Hill, New York

    MATH  Google Scholar 

  • Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 1701–1728

  • Turro E, Bochkina N, Hein AMK, Richardson S (2007) BGX: a Bioconductor package for the Bayesian integrated analysis of Affymetrix GeneChips. BMC bioinformatics 8(1):439–448

    Article  Google Scholar 

  • Vihola M (2012) Robust adaptive Metropolis algorithm with coerced acceptance rate. Stat Comput 22(5):997–1008

    Article  MathSciNet  MATH  Google Scholar 

  • Yang J (2016) Convergence and efficiency of adaptive MCMC. PhD thesis. Department of Statistical Sciences, University of Toronto. Unpublished thesis

Download references

Acknowledgements

We thank the anonymous reviewer for very helpful comments, which led to many improvements.

Author information

Authors and Affiliations

  1. Department of Statistical Sciences, University of Toronto, Toronto, Ontario, M5S 3G3, Canada

    Jeffrey S. Rosenthal & Jinyoung Yang

Authors
  1. Jeffrey S. Rosenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinyoung Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeffrey S. Rosenthal.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosenthal, J.S., Yang, J. Ergodicity of Combocontinuous Adaptive MCMC Algorithms. Methodol Comput Appl Probab 20, 535–551 (2018). https://doi.org/10.1007/s11009-017-9574-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-017-9574-3

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation