Batch means and spectral variance estimators in Markov chain Monte Carlo

Abstract

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.

Published in at http://dx.doi.org/10.1214/09-AOS735 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?

Cited by in corpus (69)

• Rank-normalization, folding, and localization: An improved $\hat{R}$ for assessing convergence of MCMC
• Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition
• Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation
• Variable transformation to obtain geometric ergodicity in the random-walk Metropolis algorithm
• Markov Chain Monte Carlo Estimation of Quantiles
• On The XUV Luminosity Evolution of TRAPPIST-1
• Statistical Inference for Model Parameters in Stochastic Gradient Descent
• Convergence analysis of the Gibbs sampler for Bayesian general linear mixed models with improper priors
• Exact sampling for intractable probability distributions via a Bernoulli factory
• Rigorous confidence bounds for MCMC under a geometric drift condition
• Importance sampling type estimators based on approximate marginal MCMC
• Bayes Shrinkage at GWAS scale: Convergence and Approximation Theory of a Scalable MCMC Algorithm for the Horseshoe Prior
• Convergence properties of Gibbs samplers for Bayesian probit regression with proper priors
• Post-Processing of MCMC
• Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo
• Computational approaches for empirical Bayes methods and Bayesian sensitivity analysis
• Diffusion approximations and control variates for MCMC
• Convergence of hybrid slice sampling via spectral gap
• A Tale of HodgeRank and Spectral Method: Target Attack Against Rank Aggregation Is the Fixed Point of Adversarial Game
• Relative fixed-width stopping rules for Markov chain Monte Carlo simulations
• Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance
• Markov Chain Monte Carlo confidence intervals
• Statistical Adaptive Stochastic Gradient Methods
• Strong Consistency of Multivariate Spectral Variance Estimators
• Estimating standard errors for importance sampling estimators with multiple Markov chains
• Path integral Monte Carlo with importance sampling for excitons interacting with an arbitrary phonon bath
• Nonasymptotic bounds on the estimation error for regenerative MCMC algorithms
• Non-asymptotic confidence intervals for MCMC in practice
• Online Covariance Matrix Estimation in Stochastic Gradient Descent
• Refinement of Thermostated Molecular Dynamics Using Backward Error Analysis
• A New Formulation for the Concerted Alchemical Calculation of van der Waals and Coulomb Components of Solvation Free Energies
• Estimating accuracy of the MCMC variance estimator: a central limit theorem for batch means estimators
• Multivariate strong invariance principles in Markov chain Monte Carlo
• Dimension-independent spectral gap of polar slice sampling
• ALAAMEE: Open-source software for fitting autologistic actor attribute models
• Limit Theorems for quadratic forms of Markov Chains
• Batch size selection for variance estimators in MCMC
• Approximating posteriors with high-dimensional nuisance parameters via integrated rotated Gaussian approximation
• Stability of Noisy Metropolis-Hastings
• Bayesian Inference for Logistic Regression Models Using Sequential Posterior Simulation
• Weighted batch means estimators in Markov chain Monte Carlo
• Estimating Monte Carlo variance from multiple Markov chains
• Slice Sampling for General Completely Random Measures
• Bayesian inference for latent factor GARCH models
• Simulation of bilayer Hamiltonians based on monitored quantum trajectories
• Analyzing MCMC Output
• On the use of Markov chain Monte Carlo methods for the sampling of mixture models
• High-Dimensional Mean-Variance Spanning Tests
• Wasserstein contraction and spectral gap of slice sampling revisited
• Quantifying Epistemic Uncertainty in Deep Learning
• Convergence Rates of Two-Component MCMC Samplers
• An MCMC Approach to Empirical Bayes Inference and Bayesian Sensitivity Analysis via Empirical Processes
• Convergence analysis of the block Gibbs sampler for Bayesian probit linear mixed models with improper priors
• Explaining the behavior of joint and marginal Monte Carlo estimators in latent variable models with independence assumptions
• Gibbs Sampling for a Bayesian Hierarchical General Linear Model
• On Size and Power of Heteroskedasticity and Autocorrelation Robust Tests
• An Empirical Comparison of Sampling Quality Metrics: A Case Study for Bayesian Nonnegative Matrix Factorization
• A Convergence Diagnostic for Bayesian Clustering
• Sliding-Window QPS (SW-QPS): A Perfect Parallel Iterative Switching Algorithm for Input-Queued Switches
• Making mean-estimation more efficient using an MCMC trace variance approach: DynaMITE
• MCMC Confidence Intervals and Biases
• Higher-Order Coverage Errors of Batching Methods via Edgeworth Expansions on $t$-Statistics
• hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty
• A Flexible Stochastic Conditional Duration Model
• Variance reduction for dependent sequences with applications to Stochastic Gradient MCMC
• Variance reduction for Markov chains with application to MCMC
• Analysis of the Polya-Gamma block Gibbs sampler for Bayesian logistic linear mixed models
• A practical sequential stopping rule for high-dimensional MCMC and its application to spatial-temporal Bayesian models
• RMCMC: A System for Updating Bayesian Models