# Variance reduction for Markov chains with application to MCMC

@article{Belomestny2020VarianceRF, title={Variance reduction for Markov chains with application to MCMC}, author={Denis Belomestny and Leonid Iosipoi and {\'E}ric Moulines and Alexey Naumov and Sergey Samsonov}, journal={Statistics and Computing}, year={2020}, volume={30}, pages={973-997} }

In this paper, we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non-asymptotic analysis of a variance reduced functional as well as by a thorough…

## 21 Citations

Variance reduction for additive functionals of Markov chains via martingale representations

- Mathematics, Computer ScienceStat. Comput.
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By rigorously analyzing the convergence properties of the proposed algorithm, it is shown that its cost-to-variance product is indeed smaller than one of the naive algorithms.

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The theoretical validity of the estimators proposed and their efficiency relative to the underlying MCMC algorithms are established and the performance and limitations of the method are illustrated.

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We introduce a general framework that constructs estimators with reduced variance for random walk Metropolis and Metropolis-adjusted Langevin algorithms. The resulting estimators require negligible…

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Postprocessing of MCMC

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- 2021

Markov chain Monte Carlo is the engine of modern Bayesian statistics, being used to approximate the posterior and derived quantities of interest. Despite this, the issue of how the output from a…

Variance Reduction for Dependent Sequences with Applications to Stochastic Gradient MCMC

- Computer Science, MathematicsSIAM/ASA J. Uncertain. Quantification
- 2021

This work proposes a novel and practical variance reduction approach for additive functionals of dependent sequences that combines the use of control variates with the minimization of randomness in these sequences.

Stein's Method Meets Computational Statistics: A Review of Some Recent Developments

- Mathematics
- 2021

Stein’s method compares probability distributions through the study of a class of linear operators called Stein operators. While mainly studied in probability and used to underpin theoretical…

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Regularizing the ordinary least squares estimator by preselecting appropriate control variates via the Lasso turns out to increase the accuracy without additional computational cost.

Vector-Valued Control Variates

- Computer Science
- 2021

Control variates are post-processing tools for Monte Carlo estimators which can lead to significant variance reduction. This approach usually requires a large number of samples, which can be…

Semi-Exact Control Functionals From Sard’s Method

- MathematicsBiometrika
- 2021

A novel control variate technique is proposed for post-processing of Markov chain Monte Carlo output, based both on Stein’s method and an approach to numerical integration due to Sard. The…

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