Multivariate output analysis for Markov chain Monte Carlo

By Dootika Vats, James M. Flegal, Galin L. Jones. Biometrika, 2015

304 citations recorded by the source corpus.

Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f97b217000687998a9358e1104dbc28933423852

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Citations in the corpus

An approximate sampler for energy-based models with divergence diagnostics (2022, background): “, 2021) require Euclidean structure for computing variances and correlations of real-valued variables (or vectors in multivariate generalizations; Vats et al., 2019; Brooks and Gelman, 1998), which are not intrinsically defined on discrete spaces only endowed with a probabilistic structure.”
Revisiting the Gelman–Rubin Diagnostic (2018, background): “A theoretically-justified lower bound on the number of effective samples required to obtain a certain level of precision has been determined for the univariate case (Gong and Flegal, 2016) and for the general multivariate problem (Vats et al., 2018a).”
Rank-Normalization, Folding, and Localization: An Improved Rˆ for Assessing Convergence of MCMC (with Discussion) (2019, methodology): “A small value of R̂ is not enough to ensure that an MCMC sample is useful in practice (Vats and Knudson, 2018). The effective sample size must also be large enough to get stable inferences for quantities of interest. Gelman et al. (2013) proposed an ESS estimate which combines autocovariance-based s”