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Variance reduction for Metropolis–Hastings samplers

Published version
Peer-reviewed

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Authors

Dellaportas, P 
Titsias, MK 

Abstract

jats:titleAbstract</jats:title>jats:pWe introduce a general framework that constructs estimators with reduced variance for random walk Metropolis and Metropolis-adjusted Langevin algorithms. The resulting estimators require negligible computational cost and are derived in a post-process manner utilising all proposal values of the Metropolis algorithms. Variance reduction is achieved by producing control variates through the approximate solution of the Poisson equation associated with the target density of the Markov chain. The proposed method is based on approximating the target density with a Gaussian and then utilising accurate solutions of the Poisson equation for the Gaussian case. This leads to an estimator that uses two key elements: (1) a control variate from the Poisson equation that contains an intractable expectation under the proposal distribution, (2) a second control variate to reduce the variance of a Monte Carlo estimate of this latter intractable expectation. Simulated data examples are used to illustrate the impressive variance reduction achieved in the Gaussian target case and the corresponding effect when target Gaussianity assumption is violated. Real data examples on Bayesian logistic regression and stochastic volatility models verify that considerable variance reduction is achieved with negligible extra computational cost.</jats:p>

Description

Keywords

OriginalPaper, Bayesian inference, Control variates, Markov chain Monte Carlo, Logistic regression, Poisson equation, Stochastic volatility

Journal Title

Statistics and Computing

Conference Name

Journal ISSN

0960-3174
1573-1375

Volume Title

Publisher

Springer Science and Business Media LLC
Sponsorship
Alan Turing Institute (TEDSA2/100056)