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Statistics and Computing

, Volume 22, Issue 5, pp 997–1008 | Cite as

Robust adaptive Metropolis algorithm with coerced acceptance rate

  • Matti Vihola
Article

Abstract

The adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen (Bernoulli 7(2):223–242, 2001) uses the estimated covariance of the target distribution in the proposal distribution. This paper introduces a new robust adaptive Metropolis algorithm estimating the shape of the target distribution and simultaneously coercing the acceptance rate. The adaptation rule is computationally simple adding no extra cost compared with the AM algorithm. The adaptation strategy can be seen as a multidimensional extension of the previously proposed method adapting the scale of the proposal distribution in order to attain a given acceptance rate. The empirical results show promising behaviour of the new algorithm in an example with Student target distribution having no finite second moment, where the AM covariance estimate is unstable. In the examples with finite second moments, the performance of the new approach seems to be competitive with the AM algorithm combined with scale adaptation.

Keywords

Acceptance rate Adaptive Markov chain Monte Carlo Ergodicity Metropolis algorithm Robustness 

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References

  1. Andrieu, C., Moulines, É.: On the ergodicity properties of some adaptive MCMC algorithms. Ann. Appl. Probab. 16(3), 1462–1505 (2006) MathSciNetMATHCrossRefGoogle Scholar
  2. Andrieu, C., Robert, C.P.: Controlled MCMC for optimal sampling. Tech. Rep. Ceremade 0125, Université Paris Dauphine (2001) Google Scholar
  3. Andrieu, C., Thoms, J.: A tutorial on adaptive MCMC. Stat. Comput. 18(4), 343–373 (2008) MathSciNetCrossRefGoogle Scholar
  4. Andrieu, C., Moulines, É., Volkov, S.: Convergence of stochastic approximation for Lyapunov stable dynamics: a proof from first principles. Technical report (2004) Google Scholar
  5. Andrieu, C., Moulines, É., Priouret, P.: Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44(1), 283–312 (2005) MathSciNetMATHCrossRefGoogle Scholar
  6. Atchadé, Y., Fort, G.: Limit theorems for some adaptive MCMC algorithms with subgeometric kernels. Bernoulli 16(1), 116–154 (2010) MathSciNetMATHCrossRefGoogle Scholar
  7. Atchadé, Y.F., Rosenthal, J.S.: On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11(5), 815–828 (2005) MathSciNetMATHCrossRefGoogle Scholar
  8. Benveniste, A., Métivier, M., Priouret, P.: Adaptive Algorithms and Stochastic Approximations. Applications of Mathematics, vol. 22. Springer, Berlin (1990) MATHGoogle Scholar
  9. Borkar, V.S.: Stochastic Approximation: A Dynamical Systems Viewpoint. Cambridge University Press, Cambridge (2008) Google Scholar
  10. Dongarra, J.J., Bunch, J.R., Moler, C.B., Stewart, G.W.: LINPACK Users’ Guide. Society for Industrial and Applied Mathematics (1979) Google Scholar
  11. Gelman, A., Roberts, G.O., Gilks, W.R.: Efficient Metropolis jumping rules. In: Bayesian Statistics 5, pp. 599–607. Oxford University Press, Oxford (1996) Google Scholar
  12. Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov Chain Monte Carlo in Practice. Chapman & Hall/CRC, Boca Raton (1998) Google Scholar
  13. Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7(2), 223–242 (2001) MathSciNetMATHCrossRefGoogle Scholar
  14. Hastie, D.: Toward automatic reversible jump Markov chain Monte Carlo. PhD thesis, University of Bristol (2005) Google Scholar
  15. Huber, P.J.: Robust Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1981) MATHCrossRefGoogle Scholar
  16. Jarner, S.F., Hansen, E.: Geometric ergodicity of Metropolis algorithms. Stoch. Process. Appl. 85, 341–361 (2000) MathSciNetMATHCrossRefGoogle Scholar
  17. Jarner, S.F., Roberts, G.O.: Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Stat. 34(4), 781–815 (2007) MathSciNetMATHGoogle Scholar
  18. Kushner, H.J., Yin, G.G.: Stochastic Approximation and Recursive Algorithms and Applications, 2nd edn. Applications of Mathematics: Stochastic Modelling and Applied Probability, vol. 35. Springer, Berlin (2003) MATHGoogle Scholar
  19. Nummelin, E.: MC’s for MCMC’ists. Int. Stat. Rev. 70(2), 215–240 (2002) MATHCrossRefGoogle Scholar
  20. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (1999) MATHGoogle Scholar
  21. Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16(4), 351–367 (2001) MathSciNetMATHCrossRefGoogle Scholar
  22. Roberts, G.O., Rosenthal, J.S.: General state space Markov chains and MCMC algorithms. Probab. Surv. 1, 20–71 (2004) MathSciNetMATHCrossRefGoogle Scholar
  23. Roberts, G.O., Rosenthal, J.S.: Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44(2), 458–475 (2007) MathSciNetMATHCrossRefGoogle Scholar
  24. Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–367 (2009) MathSciNetCrossRefGoogle Scholar
  25. Roberts, G.O., Gelman, A., Gilks, W.R.: Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7(1), 110–120 (1997) MathSciNetMATHCrossRefGoogle Scholar
  26. Saksman, E., Vihola, M.: On the ergodicity of the adaptive Metropolis algorithm on unbounded domains. Ann. Appl. Probab. 20(6), 2178–2203 (2010) MathSciNetMATHCrossRefGoogle Scholar
  27. Vihola, M.: Grapham: Graphical models with adaptive random walk Metropolis algorithms. Comput. Stat. Data Anal. 54(1), 49–54 (2010) MathSciNetCrossRefGoogle Scholar
  28. Vihola, M.: Can the adaptive Metropolis algorithm collapse without the covariance lower bound? Electron. J. Probab. 16, 45–75 (2011a) MathSciNetMATHCrossRefGoogle Scholar
  29. Vihola, M.: On the stability and ergodicity of adaptive scaling Metropolis algorithms. Preprint (2011b). arXiv:0903.4061v3

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläUniversity of JyväskyläFinland

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