Markov chain Monte Carlo: Can we trust the third significant figure?

By James M. Flegal, Murali Haran, Galin L. Jones. 2007

377 citations recorded by the source corpus.

Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/01da989f456e322a6f8d9d29e2f7a9c17d4b3b88

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In Management Science and Operations Research

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Citations in the corpus

A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo (2016, background): “A common choice of τ = 1/2 has been shown to work well in applications (Jones et al. 2006; Flegal, Haran, and Jones 2008).”
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.”
Multivariate output analysis for Markov chain Monte Carlo (2015, background): “As a consequence, estimation is reliable in the sense that if the procedure is repeated again, the estimates will not be vastly different (Flegal et al., 2008).”
Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition (2009, methodology): “In addition to generally ensuring the rapid convergence required for useful simulation, (1) is also a key sufficient condition for calculating asymptotically valid Monte Carlo standard errors (Flegal et al., 2008; Flegal and Jones, 2008; Hobert et al., 2002; Jones et al., 2006).”
The informed sampler: A discriminative approach to Bayesian inference in generative computer vision models (2014, methodology): “We use established methods for monitoring the convergence of our MCMC method [27,17].”
Revisiting the Gelman–Rubin Diagnostic (2018, methodology): “Such an improvement was indirectly implied by Flegal et al. (2008). Second, in Section 4, we present a method of selecting a principled, interpretable GR statistic threshold by identifying a one-to-one correspondence between the GR statistic and effective sample size (ESS) of estimating the mean of ”
On the Assessment of Monte Carlo Error in Simulation-Based Statistical Analyses (2009, background): “…intervals (Hall 1986; Booth and Sarkar 1998; Lee and Young 1999), evaluation of integrals (Booth and Caffo 2002; Jank and Booth 2003), and assessment of convergence of Markov chain Monte Carlo schemes in Bayesian inference (Geyer 1992; Roberts 1996; Kosorok 2000; Flegal, Haran, and Jones 2008).”
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).”
Gibbs sampling methods for Bayesian quantile regression (2011, methodology): “To assess the sampling efficiency of the proposed algorithm, we calculated Monte Carlo standard errors 26 27 and inefficiency factors for <inline-graphic xmlns:xlink="http://www.”
Geometric ergodicity of the Bayesian lasso (2013, background): “Geometric ergodicity of the Bayesian lasso 2153 Mtd is strictly positive on X (see Section 2), the same 2 + δ moment condition implies that the usual estimators of the asymptotic variance, φ(2), are consistent [2, 8, 9, 11].”
Batch means and spectral variance estimators in Markov chain Monte Carlo (2008, methodology): “We describe these approaches more fully below but the interested reader is directed to Flegal et al. (2008) and Jones et al. (2006) for more detail and comparisons with other methods.”