Batch means and spectral variance estimators in Markov chain Monte Carlo

By James M. Flegal, Galin L. Jones. 2008

271 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6518e884787ccc0b51da188171ad19dc210d4485

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Citations in the corpus

Statistical inference for model parameters in stochastic gradient descent (2016, background): “, [16, 17, 9, 14, 12, 19, 13]) is widely used for variance estimation in a time-homogeneous Markov chain.”
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].”
Revisiting the Gelman–Rubin Diagnostic (2018, background): “, 2006), spectral variance estimators and overlapping batch means estimators (Flegal and Jones, 2010), and weighted batch means estimators (Liu and Flegal, 2018). Under appropriate conditions, the estimators above are strongly consistent but are biased from below for τ 2 ∞ (Vats and Flegal, 2018). T”
Monte Carlo methods (2013, methodology): “Now, when a CLT holds, confidence intervals can be built using proper 6 approximations of σ(2)lim [10].”
A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo (2016, background): “…considerable effort establishing strongly consistent estimators for the asymptotic variance at (1) including batch means (Jones et al. 2006; Flegal and Jones 2010), spectral variance estimation (Flegal and Jones 2010), and regenerative simulation (Mykland, Tierney, and Yu 1995; Hobert et…”
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).”