Abstract
This paper proposes a family of weighted batch means variance estimators, which are computationally efficient and can be conveniently applied in practice. The focus is on Markov chain Monte Carlo simulations and estimation of the asymptotic covariance matrix in the Markov chain central limit theorem, where conditions ensuring strong consistency are provided. Finite sample performance is evaluated through auto-regressive, Bayesian spatial-temporal, and Bayesian logistic regression examples, where the new estimators show significant computational gains with a minor sacrifice in variance compared with existing methods.
Highlights
Markov chain Monte Carlo (MCMC) methods are widely used to approximate expectations with respect to a target distribution, see e.g. Liu (2001) and Robert and Casella (2004)
An MCMC simulation generates a dependent sample from the target distribution and uses ergodic averages to estimate a vector of expectations
Estimating variability is akin to estimation of the asymptotic covariance matrix in a multivariate Markov chain central limit theorem (CLT)
Summary
Markov chain Monte Carlo (MCMC) methods are widely used to approximate expectations with respect to a target distribution, see e.g. Liu (2001) and Robert and Casella (2004). BM (Flegal and Jones, 2010; Jones et al, 2006; Meketon and Schmeiser, 1984), spectral variance (SV) methods including flat top estimators (Anderson, 1994; Politis and Romano, 1995, 1996), initial sequence estimators (Geyer, 1992), recursive estimators of time-average variances (Wu et al, 2009; Yau and Chan, 2016), and regenerative simulation (Hobert et al, 2002; Mykland et al, 1995; Seila, 1982) Many of these univariate techniques can be extended to the multivariate setting, but practical challenges increase as the dimension increases. Our final example considers a Bayesian logistic regression model that illustrates weighted BM estimators with a flat top window provide more accurate coverage probabilities of multivariate confidence regions.
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