The Annals of Statistics
- Ann. Statist.
- Volume 36, Number 2 (2008), 532-554.
A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms
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 , 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 , where R is a Markov transition function on . 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.
Ann. Statist., Volume 36, Number 2 (2008), 532-554.
First available in Project Euclid: 13 March 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 62F15: Bayesian inference
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
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