Multiprocess parallel antithetic coupling for backward and forward Markov Chain Monte Carlo

Antithetic coupling is a general stratification strategy for reducing Monte Carlo variance without increasing the simulation size. The use of the antithetic principle in the Monte Carlo literature typically employs two strata via antithetic quantile coupling. We demonstrate here that further stratification, obtained by using k>2 (e.g., k=3-10) antithetically coupled variates, can offer substantial additional gain in Monte Carlo efficiency, in terms of both variance and bias. The reason for reduced bias is that antithetically coupled chains can provide a more dispersed search of the state space than multiple independent chains. The emerging area of perfect simulation provides a perfect setting for implementing the k-process parallel antithetic coupling for MCMC because, without antithetic coupling, this class of methods delivers genuine independent draws. Furthermore, antithetic backward coupling provides a very convenient theoretical tool for investigating antithetic forward coupling. However, the generation of k>2 antithetic variates that are negatively associated, that is, they preserve negative correlation under monotone transformations, and extremely antithetic, that is, they are as negatively correlated as possible, is more complicated compared to the case with k=2. In this paper, we establish a theoretical framework for investigating such issues. Among the generating methods that we compare, Latin hypercube sampling and its iterative extension appear to be general-purpose choices, making another direct link between Monte Carlo and quasi Monte Carlo.Comment: Published at http://dx.doi.org/10.1214/009053604000001075 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian methods to overcome the winner's curse in genetic studies

Parameter estimates for associated genetic variants, report ed in the initial discovery samples, are often grossly inflated compared to the values observed in the follow-up replication samples. This type of bias is a consequence of the sequential procedure in which the estimated effect of an associated genetic marker must first pass a stringent significance threshold. We propose a hierarchical Bayes method in which a spike-and-slab prior is used to account for the possibility that the significant test result may be due to chance. We examine the robustness of the method using different priors corresponding to different degrees of confidence in the testing results and propose a Bayesian model averaging procedure to combine estimates produced by different models. The Bayesian estimators yield smaller variance compared to the conditional likelihood estimator and outperform the latter in studies with low power. We investigate the performance of the method with simulations and applications to four real data examples.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS373 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Interacting multiple -- Try algorithms with different proposal distributions

We propose a new class of interacting Markov chain Monte Carlo (MCMC) algorithms designed for increasing the efficiency of a modified multiple-try Metropolis (MTM) algorithm. The extension with respect to the existing MCMC literature is twofold. The sampler proposed extends the basic MTM algorithm by allowing different proposal distributions in the multipletry generation step. We exploit the structure of the MTM algorithm with different proposal distributions to naturally introduce an interacting MTM mechanism (IMTM) that expands the class of population Monte Carlo methods and builds connections with the rapidly expanding world of adaptive MCMC. We show the validity of the algorithm and discuss the choice of the selection weights and of the different proposals. We provide numerical studies which show that the new algorithm can perform better than the basic MTM algorithm and that the interaction mechanism allows the IMTM to efficiently explore the state space.Interacting Monte Carlo, Markov chain Monte Carlo, Multiple-try Metropolis, Population Monte Carlo