Diffusion Approximations and Control Variates for MCMC

A new method is introduced for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. These control variates are obtained by minimizing the asymptotic variance associated with the Langevin diffusion over a family of functions....

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Bibliographic Details
Published inComputational mathematics and mathematical physics Vol. 64; no. 4; pp. 693 - 738
Main Authors Brosse, N., Durmus, A., Meyn, S., Moulines, E., Samsonov, S.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 2024
Springer Nature B.V
Subjects
Algorithms
Approximation
Asymptotic methods
Asymptotic properties
Bayesian analysis
Computational Mathematics and Numerical Analysis
Control systems
Diffusion
Linear control
Markov analysis
Markov chains
Mathematics
Mathematics and Statistics
Monte Carlo simulation
Optimal Control
Random walk
Statistical inference
Variance
Langevin-diffusion
control variables
MCMC algorithms
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Summary:A new method is introduced for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. These control variates are obtained by minimizing the asymptotic variance associated with the Langevin diffusion over a family of functions. To motivate our approach, we then show that the asymptotic variance of some well-known MCMC algorithms, including the Random Walk Metropolis and the (Metropolis) Unadjusted/Adjusted Langevin Algorithm, are well approximated by that of the Langevin diffusion. We finally theoretically justify the use of a class of linear control variates we introduce. In particular, we show that the variance of the resulting estimators is smaller, for a given computational complexity, than the standard Monte Carlo estimator. Several examples of Bayesian inference problems support our findings showing, in some cases, very significant reduction of the variance.
ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542524700167