Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods (original) (raw)
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\pipi-reversible Markov transition kernels PPP and QQQ. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels PiP_{i}Pi and QiQ_{i}Qi, iin0,1i\in\{0,1\}iin0,1, as soon as the kernels of each pair (P0,P1)(P_{0},P_{1})(P0,P1) and (Q0,Q1)(Q_{0},Q_{1})(Q0,Q1) 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
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
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
Vol.42 • No. 4 • August 2014