Local degeneracy of Markov chain Monte Carlo methods

Kengo Kamatani

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 713-725
  • ISSN: 1292-8100

Abstract

top
We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider 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 data augmentation (PX-DA) procedure is considered to be a remedy for it. In the sense of degeneracy, the PX-DA procedure is better than the DA procedure. However, when the number of categories is large, both procedures are degenerate and so the PX-DA procedure may not provide good estimate for the posterior distribution.

How to cite

top

Kamatani, Kengo. "Local degeneracy of Markov chain Monte Carlo methods." ESAIM: Probability and Statistics 18 (2014): 713-725. <http://eudml.org/doc/274378>.

@article{Kamatani2014,
abstract = {We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider 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 data augmentation (PX-DA) procedure is considered to be a remedy for it. In the sense of degeneracy, the PX-DA procedure is better than the DA procedure. However, when the number of categories is large, both procedures are degenerate and so the PX-DA procedure may not provide good estimate for the posterior distribution.},
author = {Kamatani, Kengo},
journal = {ESAIM: Probability and Statistics},
keywords = {Markov chain Monte Carlo; asymptotic normality; cumulative link model; numerical examples},
language = {eng},
pages = {713-725},
publisher = {EDP-Sciences},
title = {Local degeneracy of Markov chain Monte Carlo methods},
url = {http://eudml.org/doc/274378},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Kamatani, Kengo
TI - Local degeneracy of Markov chain Monte Carlo methods
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 713
EP - 725
AB - We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider 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 data augmentation (PX-DA) procedure is considered to be a remedy for it. In the sense of degeneracy, the PX-DA procedure is better than the DA procedure. However, when the number of categories is large, both procedures are degenerate and so the PX-DA procedure may not provide good estimate for the posterior distribution.
LA - eng
KW - Markov chain Monte Carlo; asymptotic normality; cumulative link model; numerical examples
UR - http://eudml.org/doc/274378
ER -

References

top
  1. [1] P. Diaconis and L. Saloff-Coste, Comparison theorems for reversible markov chains. Ann. Appl. Probab. 696 (1993). Zbl0799.60058MR1233621
  2. [2] L. Fahrmeir and H. Kaufmann, Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. Ann. Statist.13 (1985) 342–368. Zbl0594.62058MR773172
  3. [3] F.G. Foster, On the stochastic matrices associated with certain queuing processes. Ann. Math. Statist.24 (1953) 355–360. Zbl0051.10601MR56232
  4. [4] J.P. Hobert and D. Marchev, A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms. Ann. Statist.36 (2008) 532–554. Zbl1155.60031MR2396806
  5. [5] 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). Zbl1068.60002MR2053326
  6. [6] K. Kamatani, Local weak consistency of Markov chain Monte Carlo methods with application to mixture model. Bull. Inf. Cyber.45 (2013) 103–123. Zbl1294.65007MR3184420
  7. [7] K. Kamatani, Note on asymptotic properties of probit gibbs sampler. RIMS Kokyuroku1860 (2013) 140–146. 
  8. [8] K. Kamatani, Local consistency of Markov chain Monte Carlo methods. Ann. Inst. Stat. Math.66 (2014) 63–74. Zbl1281.62122MR3147545
  9. [9] J.S. Liu and C. Sabatti, Generalised Gibbs sampler and multigrid Monte Carlo for Bayesian computation. Biometrika87 (2000) 353–369. Zbl0960.65015MR1782484
  10. [10] Jun S. Liu and Ying Nian Wu, Parameter expansion for data augmentation. J. Am. Stat. Assoc.94 (1999) 1264–1274. Zbl1069.62514MR1731488
  11. [11] X.-L. Meng and David van Dyk, Seeking efficient data augmentation schemes via conditional and marginal augmentation. Biometrika 86 (1999) 301–320. Zbl1054.62505MR1705351
  12. [12] Xiao-Li Meng and David A. van Dyk, Seeking efficient data augmentation schemes via conditional and marginal augmentation. Biometrika86 (1999) 301–320. Zbl1054.62505MR1705351
  13. [13] S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Springer (1993). Zbl0925.60001MR1287609
  14. [14] Antonietta. Mira, Ordering, Slicing and Splitting Monte Carlo Markov Chains. Ph.D. thesis, University of Minnesota (1998). Zbl1127.60312MR2698214
  15. [15] P.H. Peskun, Optimum monte-carlo sampling using markov chains. Biometrika60 (1973) 607–612. Zbl0271.62041MR362823
  16. [16] G.O. Roberts and J.S. Rosenthal, General state space markov chains and mcmc algorithms. Prob. Surveys1 (2004) 20–71. Zbl1189.60131MR2095565
  17. [17] J.S. Rosenthal. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Am. Stat. Assoc.90 (1995) 558–566. Zbl0824.60077MR1340509
  18. [18] J.S. Rosenthal, Quantitative convergence rates of markov chains: A simple account. Electron. Commun. Probab.7 (2002) 123–128. Zbl1013.60053MR1917546
  19. [19] V. Roy and J.P. Hobert, 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. MR2370071
  20. [20] L. Tierney, Markov chains for exploring posterior distributions. Ann. Statist.22 (1994) 1701–1762. Zbl0829.62080MR1329166
  21. [21] L. Tierney, A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab.8 (1998) 1–9. Zbl0935.60053MR1620401
  22. [22] Wai Kong Yuen. Applications of geometric bounds to the convergence rate of Markov chains on Rn. Stoch. Process. Appl. 87 20001–23. Zbl1045.60073MR1751162

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.