Viewing the observed data of a statistical model as incomplete and augmenting its missing parts are useful for clarifying concepts and central to the invention of two well-known statistical algorithms: expectation-maximization (EM) and data augmentation. Recently, Liu, Rubin, and Wu (1998) demonstrate that expanding the parameter space along with augmenting the missing data is useful for accelerating iterative computation in an EM algorithm. The main purpose of this article is to rigorously define a parameter expanded data augmentation (PX-DA) algorithm and to study its theoretical properties. The PX-DA is a special way of using auxiliary variables to accelerate Gibbs sampling algorithms and is closely related to reparameterization techniques. Theoretical results concerning the convergence rate of the PX-DA algorithm and the choice of prior for the expansion parameter are obtained. In order to understand the role of the expansion parameter, we establish a new theory for iterative condi...

“Extended Ensemble Monte Carlo ” is a generic term that indicates a set of algorithms which are now popular in a variety of fields in physics and statistical information processing. Exchange Monte Carlo (Metropolis-Coupled Chain, Parallel Tempering), Simulated Tempering (Expanded Ensemble Monte Carlo), and Multicanonical Monte Carlo (Adaptive Umbrella Sampling) are typical members of this family. Here we give a cross-disciplinary survey of these algorithms with special emphasis on the great flexibility of the underlying idea. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Sec. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. In Sec. 6, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 7. We stress

: Perfect simulation algorithms based on Propp and Wilson (1996) have so far been of limited use for sampling problems of interest in statistics. We specify a new family of perfect sampling algorithms obtained by combining MCMC tempering algorithms with dominated coupling from the past, and demonstrate that our algorithms will be useful for sample based inference. Perfect tempering algorithms are less efficient than the MCMC algorithms on which they typically depend. However, samples returned by perfect tempering are distributed according to the intended distribution, so that these new sampling algorithms do not suffer from the convergence problems of MCMC. Perfect tempering is related to rejection sampling. When rejection sampling has been tried, but has proved impractical, it may be possible to convert the rejection algorithm into a perfect tempering algorithm, with a significant gain in algorithm efficiency. Keywords: Bayesian inference; Dominated coupling from the past; Exact samp...

... this article, we present a generalized version of the Gibbs sampler that allows flexible conditional moves defined by groups of transformations. We explore its connection with the multigrid Monte Carlo (MGMC) of Goodman & Sokal and propose its use in designing more efficient samplers. The generalized Gibbs sampler provides a framework encompassing a class of recently proposed tricks, such as parameter expansion, reparameterization, blocking and grouping. To illustrate, this new method is applied to Bayesian inference problems for the multivariate Gaussian, nonlinear state-space models, ordinal data, and stochastic differential equations with discrete observations.

this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.

This article provides a simple method to accelerate Markov chain Monte Carlo sampling algorithms, such as the data augmentation algorithm and the Gibbs sampler, via alternating subspace-spanning resampling (ASSR). The ASSR algorithm often shares the simplicity of its parent sampler but has dramatically improved efficiency. The methodology is illustrated with Bayesian estimation for analysis of censored data from fractionated experiments. The relationships between ASSR and existing methods are also discussed.

Abstract not found

We generalize to general state spaces a Monte Carlo algorithm recently proposed by Wang and Landau (2001). The algorithm can be seen as an adaptive Markov Chain Monte Carlo algorithm where a partition of the state space is chosen and the target density sequentially reweighted is each component of the partition in such a way that at the limit the empirical occupation measure of the components are equal. We develop convergence results for the algorithm. We also show that the algorithm can be used to e ciently self-tune the weights (the so-called pseudo prior) in the simulated tempering. Applied to the reversible jump, the Wang-Landau algorithm can also be used to improve on the between-model acceptance rate of the algorithm. Simulations example are given to illustrate.

AdaptivePolar Sampling #APS# is proposed as a Markovchain Monte Carlo method for Bayesian analysis of models with ill-behaved posterior distributions. In order to sample e#ciently from such a distribution, a location-scale transformation and a transformation to polar coordinates are used. After the transformation to polar coordinates, a MetropolisHastings algorithm is applied to sample directions and, conditionally on these, distances are generated byinverting the CDF. A sequential procedure is applied to update the location and scale.

We previously developed an algorithm, called resolution exchange, which improves canonical sampling of atomic resolution models by swapping conformations between high- and low-resolution simulations. 1 Here, we demonstrate a generally applicable incremental coarsening procedure and apply the algorithm to a larger peptide, met-enkephalin. In addition, we demonstrate a combination of resolution and temperature exchange, in which the coarser simulations are also at elevated temperatures. Both simulations are implemented in a “top-down” mode, to allow efficient allocation of CPU time among the different replicas. Atomic resolution simulations of proteins are currently limited to short durations (less than one µsec) 2 or small systems (less than 100 residues). 3,4 Furthermore, accurate calculations involving large conformational changes are not possible for any system, as the cost of calculating entropic contributions is too great. Indeed, the cost of such calculations is only going to increase, as empirical force fields are improved by including polarization effects, either in a classical way 5,6 or in a semiclassical way. 7 Thoroughly sampling the space of conformations is essential for a number of problems. From a purely biological perspective, there is a growing awareness of the importance of protein fluctuations—over and above the static picture—in the function of most proteins. 8 Allostery and conformational changes dramatic enough to be captured experimentally are just two examples of the existence of such fluctuations. 9,10 In a computational context, careful validation of empirical forcefields requires confidence