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Local degeneracy of Markov chain Monte Carlo methods∗

Published online by Cambridge University Press:  22 October 2014

Kengo Kamatani
Kengo Kamatani*
Affiliation:
Graduate School of Engineering Science, Osaka University, Machikaneyama-cho 1-3, Toyonaka-si, 560-0043 Osaka, Japan. [email protected]
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Abstract

We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes theperformances of MCMC procedures are poor and there are great importance for the study ofsuch behavior. In this paper we call degeneracy for a particular type of poorperformances. We show some equivalent conditions for degeneracy. As an application, weconsider the cumulative probit model. It is well known that the natural data augmentation(DA) procedure does not work well for this model and the so-called parameter-expanded dataaugmentation (PX-DA) procedure is considered to be a remedy for it. In the sense ofdegeneracy, the PX-DA procedure is better than the DA procedure. However, when the numberof categories is large, both procedures are degenerate and so the PX-DA procedure may notprovide good estimate for the posterior distribution.

Keywords

Markov chain Monte Carloasymptotic normalitycumulative link model
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 713 - 725
Copyright
© EDP Sciences, SMAI 2014

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References

P. Diaconis and L. Saloff-Coste, Comparison theorems for reversible markov chains. Ann. Appl. Probab. 696 (1993).
Fahrmeir, L. and Kaufmann, H., Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. Ann. Statist. 13 (1985) 342–368. Google Scholar
Foster, F.G., On the stochastic matrices associated with certain queuing processes. Ann. Math. Statist. 24 (1953) 355–360. Google Scholar
Hobert, J.P. and Marchev, D., A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms. Ann. Statist. 36 (2008) 532–554. Google Scholar
K. Itô, Stochastic processes. ISBN 3-540-20482-2. Lectures given at Aarhus University, Reprint of the 1969 original, edited and with a foreword by Ole E. Barndorff-Nielsen and Ken-iti Sato. Springer-Verlag, Berlin (2004).
Kamatani, K., Local weak consistency of Markov chain Monte Carlo methods with application to mixture model. Bull. Inf. Cyber. 45 (2013) 103–123. Google Scholar
Kamatani, K., Note on asymptotic properties of probit gibbs sampler. RIMS Kokyuroku 1860 (2013) 140–146. Google Scholar
Kamatani, K., Local consistency of Markov chain Monte Carlo methods. Ann. Inst. Stat. Math. 66 (2014) 63–74. Google Scholar
Liu, J.S. and Sabatti, C., Generalised Gibbs sampler and multigrid Monte Carlo for Bayesian computation. Biometrika 87 (2000) 353–369. Google Scholar
Liu, Jun S. and Wu, Ying Nian, Parameter expansion for data augmentation. J. Am. Stat. Assoc. 94 (1999) 1264–1274. Google Scholar
Meng, X.-L. and van Dyk, David, Seeking efficient data augmentation schemes via conditional and marginal augmentation. Biometrika 86 (1999) 301–320. Google Scholar
Meng, Xiao-Li and van Dyk, David A., Seeking efficient data augmentation schemes via conditional and marginal augmentation. Biometrika 86 (1999) 301–320. Google Scholar
S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Springer (1993).
Antonietta. Mira, Ordering, Slicing and Splitting Monte Carlo Markov Chains. Ph.D. thesis, University of Minnesota (1998).
Peskun, P.H., Optimum monte-carlo sampling using markov chains. Biometrika 60 (1973) 607–612. Google Scholar
Roberts, G.O. and Rosenthal, J.S., General state space markov chains and mcmc algorithms. Prob. Surveys 1 (2004) 20–71. Google Scholar
Rosenthal, J.S.. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Am. Stat. Assoc. 90 (1995) 558–566. Google Scholar
Rosenthal, J.S., Quantitative convergence rates of markov chains: A simple account. Electron. Commun. Probab. 7 (2002) 123–128. Google Scholar
Roy, V. and Hobert, J.P., Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 (2007) 607–623. Google Scholar
Tierney, L., Markov chains for exploring posterior distributions. Ann. Statist. 22 (1994) 1701–1762. Google Scholar
Tierney, L., A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab. 8 (1998) 1–9. Google Scholar
Yuen, Wai Kong. Applications of geometric bounds to the convergence rate of Markov chains on Rn. Stoch. Process. Appl. 87 20001–23.