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April 2008 A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms
James P. Hobert, Dobrin Marchev
Ann. Statist. 36(2): 532-554 (April 2008). DOI: 10.1214/009053607000000569


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.


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James P. Hobert. Dobrin Marchev. "A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms." Ann. Statist. 36 (2) 532 - 554, April 2008.


Published: April 2008
First available in Project Euclid: 13 March 2008

zbMATH: 1155.60031
MathSciNet: MR2396806
Digital Object Identifier: 10.1214/009053607000000569

Primary: 60J27
Secondary: 62F15

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

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 2 • April 2008
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