The Annals of Statistics

A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms

James P. Hobert and Dobrin Marchev

Full-text: Open access

Abstract

The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo (MCMC) algorithm that is based on a Markov transition density of the form $p(x|x')=\int_{\mathsf{Y}}f_{X|Y}(x|y)f_{Y|X}(y|x')\,dy$, where fX|Y and fY|X are conditional densities. The PX-DA and marginal augmentation algorithms of Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264–1274] and Meng and van Dyk [Biometrika 86 (1999) 301–320] are alternatives to DA that often converge much faster and are only slightly more computationally demanding. The transition densities of these alternative algorithms can be written in the form $p_{R}(x|x')=\int_{\mathsf{Y}}\int _{\mathsf{Y}}f_{X|Y}(x|y')R(y,dy')f_{Y|X}(y|x')\,dy$, where R is a Markov transition function on $\mathsf{Y}$. We prove that when R satisfies certain conditions, the MCMC algorithm driven by pR is at least as good as that driven by p in terms of performance in the central limit theorem and in the operator norm sense. These results are brought to bear on a theoretical comparison of the DA, PX-DA and marginal augmentation algorithms. Our focus is on situations where the group structure exploited by Liu and Wu is available. We show that the PX-DA algorithm based on Haar measure is at least as good as any PX-DA algorithm constructed using a proper prior on the group.

Article information

Source
Ann. Statist., Volume 36, Number 2 (2008), 532-554.

Dates
First available in Project Euclid: 13 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1205420510

Digital Object Identifier
doi:10.1214/009053607000000569

Mathematical Reviews number (MathSciNet)
MR2396806

Zentralblatt MATH identifier
1155.60031

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 62F15: Bayesian inference

Keywords
Central limit theorem convergence rate group action left-Haar measure Markov chain Markov operator Monte Carlo nonpositive recurrent operator norm relatively invariant measure topological group

Citation

Hobert, James P.; Marchev, Dobrin. A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms. Ann. Statist. 36 (2008), no. 2, 532--554. doi:10.1214/009053607000000569. https://projecteuclid.org/euclid.aos/1205420510


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