Fixed-Width Output Analysis for Markov Chain Monte Carlo

By Galin L. Jones, Murali Haran, Brian C. Caffo, R. Neath. 2006

363 citations recorded by the source corpus.

Markov chain Monte Carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages. A fundamental question is when sampling should stop; that is, at what point the ergodic averages are good estimates of the desired quantities. We consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user-specified value. Hence calculating a Monte Carlo standard error is a critical step in assessing the simulation output. We consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We give sufficient conditions for the strong consistency of both methods and investigate their finite-sample properties in various examples.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4123245eb03012bc17fa7e96d528bc7e4451d854

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Citations in the corpus

Geometric ergodicity of the Bayesian lasso (2013, background): “Moreover, because the Mtd is strictly positive on X (see Section 2), the same 2 + moment condition implies that the usual estimators of the asymptotic variance, 2, are consistent [2, 8, 9, 11].”
Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression (2007, background): “We now have that E[E[V.β′/|z, y]|β, y] E[p+ zTz|β, y]=p+ n∑ i=1 E[z2i |β, y]: Standard results for the truncated normal distribution (see, for example, Johnson and Kotz (1970)) imply that, if U ∼TN.ξ, 1, 1/, then E[U2]=1+ ξ2 + ξ φ.ξ/ Φ.ξ/ , where φ.·/ with only a single argument denotes the…”
Dimension reduction and alleviation of confounding for spatial generalized linear mixed models (2010, methodology): “To ensure that the Monte Carlo standard errors were sufficiently small, we simulated sample paths of length 2 million in all cases (Flegal et al., 2008; Jones et al., 2006).”
Bayesian Sparse Group Selection (2016, methodology): “To assess the convergence of our MCMC algorithm, the Monte Carlo standard error (MCSE) in Jones et al. (2006) and Flegal, Haran, and Jones (2008) can be used. Since the selection results are determined based on the marginal probabilities of the indicators, the MCSE of the posterior probability, P (γ”
Multivariate output analysis for Markov chain Monte Carlo (2015, background): “problems (Jones et al., 2006; Flegal and Jones, 2010) and for steady state simulations (Damerdji, 1994; Glynn and Iglehart, 1990; Glynn and Whitt, 1991).”
Revisiting the Gelman–Rubin Diagnostic (2018, neutral): “This includes batch means estimators and regenerative estimators (Jones et al., 2006), spectral variance estimators and overlapping batch means estimators (Flegal and Jones, 2010), and weighted batch means estimators (Liu and Flegal, 2018).”
A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo (2016, methodology): “Jones et al. (2006) and Flegal, Haran, and Jones (2008) showed that the simple FWSR is superior to using convergence diagnostics as stopping criteria.”