Information bounds for Gibbs samplers
Priscilla E. Greenwood, Ian W. McKeague, and Wolfgang Wefelmeyer
Source: Ann. Statist. Volume 26, Number 6 (1998), 2128-2156.
Abstract
If we wish to estimate efficiently the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estimators that use no information about the underlying distribution. The empirical estimator has noticeably smaller variance for deterministic sweep. The variance bound for random sweep is in general smaller than for deterministic sweep, but the two are equal if the target distribution is continuous. If the components of the target distribution are not strongly dependent, the empirical estimator is close to efficient under deterministic sweep, and its asymptotic variance approximately doubles under random sweep.
Primary Subjects: 62M05, 62G20
Keywords: Efficient estimator; empirical estimator; Markov chain Monte Carlo; variance bound
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1024691464
Mathematical Reviews number (MathSciNet):
MR1700224
Digital Object Identifier: doi:10.1214/aos/1024691464
Zentralblatt MATH identifier:
0927.62080
References
Besag, J. and Kooperberg, C. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733-746.
Mathematical Reviews (MathSciNet):
MR97b:62164
Zentralblatt MATH:
0899.62123
Bickel, P. J. (1993). Estimation in semiparametric models. In Multivariate Analy sis: Future Directions (C. R. Rao, ed.) 55-73. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet):
MR94m:62008
Casella, G. and Robert, C. P. (1996). Rao-Blackwell theorem for sampling schemes with rejection. C. R. Acad. Sci. Paris S´er. I Math. 322 571-576.
Mathematical Reviews (MathSciNet):
MR97b:62036
Clifford, P. and Nicholls, G. (1995). A Metropolis sampler for poly gonal image reconstruction. Technical report, Dept. Statistics, Oxford Univ.
Fishman, G. (1996). Coordinate selection rules for Gibbs sampling. Ann. Appl. Probab. 6 444-465.
Mathematical Reviews (MathSciNet):
MR97k:62058
Zentralblatt MATH:
0855.60060
Frigessi, A., Hwang, C.-R., Sheu, S. J. and Di Stefano, P. (1993). Convergence rates of the Gibbs sampler, the Metropolis algorithm, and other single-site updating dy namics. J. Roy. Statist. Soc. Ser. B 55 205-220.
Frigessi, A., Hwang, C.-R. and Younes, L. (1992). Optimal spectral structure of reversible stochastic matrices, Monte Carlo methods and the simulation of Markov random fields. Ann. Appl. Probab. 2 610-628.
Mathematical Reviews (MathSciNet):
MR93e:60133
Zentralblatt MATH:
0756.60057
Frigessi, A. and Rue, H. (1998). Antithetic coupling of two Gibbs sampler chains. Preprint 7/98, Dept. Mathematical Sciences, Norwegian Univ. of Technology and Science, Trondheim.
Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398-409.
Mathematical Reviews (MathSciNet):
MR92k:62087
Zentralblatt MATH:
0702.62020
Gelfand, A. E. and Smith, A. F. M. (1991). Gibbs sampling for marginal posterior expectations. Comm. Statist. Theory Methods 20 1747-1766.
Mathematical Reviews (MathSciNet):
MR92j:62041
Gey er, C. J. (1992). Practical Markov chain Monte Carlo. Statist. Sci. 7 473-483.
Gey er, C. J. (1995). Conditioning in Markov chain Monte Carlo. J. Comput. Graph. Statist. 4 148-154.
Mathematical Reviews (MathSciNet):
MR96g:62037
Green, P. J. and Han, X.-L. (1992). Metropolis methods, Gaussian proposals and antithetic variables. Stochastic Models, Statistical Methods, and Algorithms in Image Analy sis Lecture Notes in Statist. 74 142-164. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1188484
Greenwood, P. E., McKeague, I. W. and Wefelmey er, W. (1996). Outperforming the Gibbs sampler empirical estimator for nearest neighbor random fields. Ann. Statist. 24 1433- 1456.
Mathematical Reviews (MathSciNet):
MR97k:65021
Zentralblatt MATH:
0871.62083
Greenwood, P. E., McKeague, I. W. and Wefelmey er, W. (1998). Von Mises ty pe statistics for single site updated local interaction random fields. Statist. Sinica. To appear.
Greenwood, P. E. and Wefelmey er, W. (1990). Efficiency of estimators for partially specified filtered models. Stochastic Process. Appl. 36 353-370.
Mathematical Reviews (MathSciNet):
MR92d:62118
Zentralblatt MATH:
0719.62050
Greenwood, P. E. and Wefelmey er, W. (1995). Efficiency of empirical estimators for Markov chains. Ann. Statist. 23 132-143.
Mathematical Reviews (MathSciNet):
MR96m:62151
Zentralblatt MATH:
0822.62067
Greenwood, P. E. and Wefelmey er, W. (1998). Reversible Markov chains and optimality of sy mmetrized empirical estimators. Bernoulli. To appear.
H´ajek, J. (1970). A characterization of limiting distributions of regular estimates. Z. Wahrsch. Verw. Gebiete 14 323-330.
Mathematical Reviews (MathSciNet):
MR44:1141
Higdon, D. M. (1998). Auxiliary variable methods for Markov chain Monte Carlo with applications. J. Amer. Statist. Assoc. 93 585-595.
Zentralblatt MATH:
0953.62103
H ¨opfner, R. (1993). On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models. Probab. Theory Related Fields 94 375-398.
Mathematical Reviews (MathSciNet):
MR94d:62209
H ¨opfner, R., Jacod, J. and Ladelli, L. (1990). Local asy mptotic normality and mixed normality for Markov statistical models. Probab. Theory Related Fields 86 105-129.
Ingrassia, S. (1994). On the rate of convergence of the Metropolis algorithm and Gibbs sampler by geometric bounds. Ann. Appl. Probab. 4 347-389.
Mathematical Reviews (MathSciNet):
MR95d:60114
Zentralblatt MATH:
0802.60061
Kessler, M., Schick, A. and Wefelmey er, W. (1998). The information in the marginal law of a Markov chain. Unpublished manuscript.
Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phy s. 104 1-19.
Mathematical Reviews (MathSciNet):
MR87i:60038
Zentralblatt MATH:
0588.60058
Liu, J. S. (1996). Metropolized Gibbs sampler: an improvement. Technical report, Dept. Statistics, Stanford Univ.
Liu, J. S., Wong, W. H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika 81 27-40.
Mathematical Reviews (MathSciNet):
MR95d:62133
Zentralblatt MATH:
0811.62080
Liu, J. S., Wong, W. H. and Kong, A. (1995). Covariance structure and convergence rate of the Gibbs sampler with various scans. J. Roy. Statist. Soc. Ser. B 57 157-169.
Mathematical Reviews (MathSciNet):
MR96i:62022
Zentralblatt MATH:
0811.60056
McEachern, S. N. and Berliner, L. M. (1994). Subsampling the Gibbs sampler. Amer. Statist. 48 188-190.
McKeague, I. W. and Wefelmey er, W. (1998). Markov chain Monte Carlo and Rao-Blackwellization. J. Statist. Plann. Inference. To appear.
Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101-121.
Mathematical Reviews (MathSciNet):
MR98c:60081
Zentralblatt MATH:
0854.60065
Mey n, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
Mathematical Reviews (MathSciNet):
MR95j:60103
Mey n, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981-1011.
Mathematical Reviews (MathSciNet):
MR95j:60106
Penev, S. (1991). Efficient estimation of the stationary distribution for exponentially ergodic Markov chains. J. Statist. Plann. Inference 27 105-123.
Mathematical Reviews (MathSciNet):
MR92d:62114
Zentralblatt MATH:
0727.62079
Peskun, P. H. (1973). Optimum Monte Carlo sampling using Markov chains. Biometrika 60 607- 612.
Mathematical Reviews (MathSciNet):
MR50:15261
Zentralblatt MATH:
0271.62041
Raftery, A. E. and Lewis, S. M. (1992). How many iterations in the Gibbs sampler? In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid, A. F. M. Smith, eds.) 763-773. Oxford Univ. Press.
Mathematical Reviews (MathSciNet):
MR2002g:47027
Roberts, G. O. and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. J. Roy. Statist. Soc. Ser. B 56 377-384.
Mathematical Reviews (MathSciNet):
MR95f:65023
Zentralblatt MATH:
0796.62029
Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hy brid Markov chains. Electron. Comm. Probab. 2 13-25.
Roberts, G. O. and Sahu, S. K. (1997). Updating schemes, correlation structure, blocking and parameterisation for the Gibbs sampler. J. Roy al. Statist. Soc. Ser. B 59 291-317.
Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multivariate Hastings and Metropolis algorithms. Biometrika 83 95-110.
Rosenthal, J. S. (1995). Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90 558-566.
Mathematical Reviews (MathSciNet):
MR96e:62167b
Zentralblatt MATH:
0824.60077
Schmeiser, B. and Chen, M. H. (1991). On random-direction Monte Carlo sampling for evaluating multidimensional integrals. Technical report, Dept. Statistics, Purdue Univ.
Spiegelhalter, D. J., Thomas, A. and Best, N. G. (1996). Computation on Bayesian graphical models. In Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 407-426. Oxford Univ. Press.
Mathematical Reviews (MathSciNet):
MR97g:62053
Swendsen, R. H. and Wang, J.-S. (1987). Nonuniversal critical dy namics in Monte Carlo simulations. Phy s. Rev. Lett. 58 86-88.
Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation. J. Amer. Statist. Assoc. 82 528-540.
Mathematical Reviews (MathSciNet):
MR88h:62050
Zentralblatt MATH:
0619.62029
The Annals of Statistics
