The Annals of Statistics

Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods

Florian Maire, Randal Douc, and Jimmy Olsson

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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.

Article information

Source
Ann. Statist., Volume 42, Number 4 (2014), 1483-1510.

Dates
First available in Project Euclid: 7 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1407420006

Digital Object Identifier
doi:10.1214/14-AOS1209

Mathematical Reviews number (MathSciNet)
MR3262458

Zentralblatt MATH identifier
1319.60152

Subjects
Primary: 60J22: Computational methods in Markov chains [See also 65C40] 65C05: Monte Carlo methods
Secondary: 62J10: Analysis of variance and covariance

Keywords
Markov chain Monte Carlo asymptotic variance Peskun ordering inhomogeneous Markov chains pseudo-marginal algorithms

Citation

Maire, Florian; Douc, Randal; Olsson, Jimmy. Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods. Ann. Statist. 42 (2014), no. 4, 1483--1510. doi:10.1214/14-AOS1209. https://projecteuclid.org/euclid.aos/1407420006


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