Fixed-Width Output Analysis for Markov Chain Monte Carlo

Abstract

Markov chain Monte Carlo is a method of producing a correlated sample in order to estimate features of a complicated target distribution via simple ergodic averages. A fundamental question in MCMC applications is when should the sampling stop? That is, when are the ergodic averages good estimates of the desired quantities? We consider a method that stops the MCMC sampling the first time the width of a confidence interval based on the ergodic averages is less than a user-specified value. Hence calculating Monte Carlo standard errors is a critical step in assessing the output of the simulation. In particular, we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We describe sufficient conditions for the strong consistency and asymptotic normality of both methods and investigate their finite sample properties in a variety of examples.

A Bayesian hierarchical framework for spatial modeling of fMRI data

“…The volume of data in question prevents formal use of MCMC convergence tools, such as consistent batch means (Jones et al, 2006). Instead, more informal approaches were taken.…”

Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors

“…Consequently, both Markov operators are compact, which implies that both Markov chains are geometrically ergodic. This is very important from a practical standpoint because geometric ergodicity guarantees the existence of the central limit theorems that form the basis of all the standard methods of calculating valid asymptotic standard errors for MCMC-based estimators (see, e.g., Jones, Haran, Caffo and Neath, 2006). Moreover, our results imply that for each i ∈ N, the ith largest eigenvalue of the sandwich operator is less than or equal to the corresponding eigenvalue of the DA operator, with strict inequality for at least one i.…”

Geometric ergodicity of Gibbs samplers for Bayesian general linear mixed models with proper priors

“…In particular, when the chain is geometric, sample averages satisfy central limit theorems, and these allow for the computation of asymptotically valid standard errors for MCMC-based estimates (Bednorz and Łatuszyński, 2007;Flegal and Jones, 2010;Hobert et al, 2002;Jones et al, 2006). The ability to compute such standard errors is very important from a practical standpoint because it leads to a coherent strategy for deciding how long to run the simulation.…”