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

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 pwhere 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 pR(x|xWe 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.

Sandwich algorithms for Bayesian variable selection

“…Note that P G is a Haar algorithm, and hence can be represented as a DA algorithm [14,Theorem 4]. Further, any DA algorithm is positive [18,Lemma 3.2], so P G is positive.…”

“…Using the terminology in Yu and Meng [27] and Khare and Hobert [16], we call the corresponding algorithm the sandwich algorithm with sandwich move S. It was shown in Hobert and Marchev [14] that any sandwich algorithm is valid in the sense that the reversibility of S with respect to π Y implies the reversibility of P S with respect to π X . …”

“…If there exists a group structure G that acts on Y, then any probability distribution r supported on G would correspond to a PXDA algorithm, and we denote its X-chain operator by P r G . It turns out that, with a fixed group G, the entire class of PXDA algorithm {P r G } r is dominated by an algorithm called the Haar PXDA, or simply the Haar algorithm, on G. Specifically, let P G denote the operator of the X-chain of the Haar algorithm, then Hobert and Marchev [14] has shown that P ≥ P r G ≥ P G and P ≤ E P r G ≤ E P G for any r supported on G. Further, the Haar costs no more than the PXDA per iteration, so we will restrict our attention to Haar algorithms only. A description of PXDA and its relationship to the Haar algorithm can be found in Section 3 of the online supplement.…”

“…Our proof only requires elementary knowledge of discrete groups.) Since ν G ′ is supported in G ′ ⊆ G, it is also a probability distribution on G. Hence, P G ′ can be considered a PXDA algorithm associated with group G. Then P G is better than P G ′ in both orderings according to Hobert and Marchev [14].…”

“…There is an extensive literature on using the sandwich method [19,20,14] to improve the convergence rate of DA algorithms. However, such methods have not been successfully utilized for Bayesian variable selection.…”