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August 2014 Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods
Florian Maire, Randal Douc, Jimmy Olsson
Ann. Statist. 42(4): 1483-1510 (August 2014). DOI: 10.1214/14-AOS1209

Abstract

In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $\pi$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_{i}$ and $Q_{i}$, $i\in\{0,1\}$, as soon as the kernels of each pair $(P_{0},P_{1})$ and $(Q_{0},Q_{1})$ can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-augmentation-type Metropolis–Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis–Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm.

Citation

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Florian Maire. Randal Douc. Jimmy Olsson. "Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods." Ann. Statist. 42 (4) 1483 - 1510, August 2014. https://doi.org/10.1214/14-AOS1209

Information

Published: August 2014
First available in Project Euclid: 7 August 2014

zbMATH: 1319.60152
MathSciNet: MR3262458
Digital Object Identifier: 10.1214/14-AOS1209

Subjects:
Primary: 60J22 , 65C05
Secondary: 62J10

Keywords: asymptotic variance , Inhomogeneous Markov chains , Markov chain Monte Carlo , Peskun ordering , pseudo-marginal algorithms

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 4 • August 2014
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