Batch means and spectral variance estimators in Markov chain Monte Carlo

Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.

[1]  R. Fisher.  The Advanced Theory of Statistics , 1943, Nature.

[2]  J. Doob.  Stochastic processes , 1953 .

[3]  M. Kendall,et al.  The Advanced Theory of Statistics: Vol. I—Distribution Theory , 1959 .

[4]  P. Whittle,et al.  Bounds for the Moments of Linear and Quadratic Forms in Independent Variables , 1960 .

[5]  P. Major,et al.  An approximation of partial sums of independent RV'-s, and the sample DF. I , 1975 .

[6]  P. Major,et al.  An approximation of partial sums of independent RV's, and the sample DF. II , 1975 .

[7]  W. Philipp,et al.  Almost sure invariance principles for partial sums of weakly dependent random variables , 1975 .

[8]  Péter Major,et al.  The approximation of partial sums of independent RV's , 1976 .

[9]  Linus Schrage,et al.  A guide to simulation , 1983 .

[10]  Paul Bratley,et al.  A guide to simulation , 1983 .

[11]  Bruce W. Schmeiser,et al.  Overlapping batch means: something for nothing? , 1984, WSC '84.

[12]  S. Singh.  Nonlinear Functional Analysis and Its Applications , 1986 .

[13]  Peter D. Welch,et al.  On the relationship between batch means, overlapping means and spectral estimation , 1987, WSC '87.

[14]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

[15]  Donald L. Iglehart,et al.  Simulation Output Analysis Using Standardized Time Series , 1990, Math. Oper. Res..

[16]  Ward Whitt,et al.  Estimating the asymptotic variance with batch means , 1991, Oper. Res. Lett..

[17]  H. Damerdji,et al.  Strong consistency and other properties of the spectral variance estimator , 1991 .

[18]  P. Glynn,et al.  The Asymptotic Validity of Sequential Stopping Rules for Stochastic Simulations , 1992 .

[19]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[20]  S. Meyn,et al.  Computable Bounds for Geometric Convergence Rates of Markov Chains , 1994 .

[21]  Halim Damerdji,et al.  Strong Consistency of the Variance Estimator in Steady-State Simulation Output Analysis , 1994, Math. Oper. Res..

[22]  Nicholas G. Polson,et al.  On the Geometric Convergence of the Gibbs Sampler , 1994 .

[23]  J. Rosenthal.  Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo , 1995 .

[24]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[25]  B. Schmeiser,et al.  Optimal mean-squared-error batch sizes , 1995 .

[26]  Halim Damerdji,et al.  Mean-Square Consistency of the Variance Estimator in Steady-State Simulation Output Analysis , 1995, Oper. Res..

[27]  Endre Csáki,et al.  On additive functionals of Markov chains , 1995 .

[28]  Bin Yu,et al.  Regeneration in Markov chain samplers , 1995 .

[29]  R. Tweedie,et al.  Rates of convergence of the Hastings and Metropolis algorithms , 1996 .

[30]  Jeffrey S. Rosenthal,et al.  Analysis of the Gibbs Sampler for a Model Related to James-stein Estimators , 2007 .

[31]  David Goldsman,et al.  Large-Sample Results for Batch Means , 1997 .

[32]  C. Geyer,et al.  Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model , 1998 .

[33]  G. Roberts,et al.  Adaptive Markov Chain Monte Carlo through Regeneration , 1998 .

[34]  J. Rosenthal,et al.  Convergence of Slice Sampler Markov Chains , 1999 .

[35]  Jun S. Liu,et al.  Parameter Expansion for Data Augmentation , 1999 .

[36]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[37]  S. F. Jarner,et al.  Geometric ergodicity of Metropolis algorithms , 2000 .

[38]  Xiao-Li Meng,et al.  The Art of Data Augmentation , 2001 .

[39]  Galin L. Jones,et al.  Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo , 2001 .

[40]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[41]  Galin L. Jones,et al.  On the applicability of regenerative simulation in Markov chain Monte Carlo , 2002 .

[42]  J. Rosenthal,et al.  General state space Markov chains and MCMC algorithms , 2004, math/0404033.

[43]  J. Hobert,et al.  Geometric Ergodicity of van Dyk and Meng's Algorithm for the Multivariate Student's t Model , 2004 .

[44]  Galin L. Jones.  On the Markov chain central limit theorem , 2004, math/0409112.

[45]  Galin L. Jones,et al.  Sufficient burn-in for Gibbs samplers for a hierarchical random effects model , 2004, math/0406454.

[46]  Olle Häggström,et al.  On the central limit theorem for geometrically ergodic Markov chains , 2005 .

[47]  D. Hinkley.  Annals of Statistics , 2006 .

[48]  Galin L. Jones,et al.  Fixed-Width Output Analysis for Markov Chain Monte Carlo , 2006, math/0601446.

[49]  Alicia A. Johnson,et al.  Gibbs Sampling for a Bayesian Hierarchical Version of the General Linear Mixed Model , 2007 .

[50]  J. Hobert,et al.  Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression , 2007 .

[51]  A Few Remarks on “Fixed-Width Output Analysis for Markov Chain Monte Carlo” by Jones et al , 2007 .

[52]  Murali Haran,et al.  Markov chain Monte Carlo: Can we trust the third significant figure? , 2007, math/0703746.

[53]  J. Hobert,et al.  Block Gibbs Sampling for Bayesian Random Effects Models With Improper Priors: Convergence and Regeneration , 2009 .

[54]  Variable-at-a-time Implementations of Metropolis-Hastings , 2009 .

[55]  T. Cipra.  Statistical Analysis of Time Series , 2010 .

[56]  Murray D. Burke,et al.  Strong Approximations in Probability and Statistics , 2011, International Encyclopedia of Statistical Science.

[57]  Charles J. Geyer,et al.  Likelihood inference for spatial point processes , 2019, Stochastic Geometry.