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

Hobert, James P. ; Marchev, Dobrin

Ann. Statist., Tome 36 (2008) no. 1, p. 532-554 / Harvested from Project Euclid

Résumé

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 f_X|Y and f_Y|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 p_R 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.

Détail

Publié le : 2008-04-15
Classification:  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,  60J27,  62F15