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Journal ArticleDOI

The pseudo-marginal approach for efficient Monte Carlo computations

01 Apr 2009-Annals of Statistics (Institute of Mathematical Statistics)-Vol. 37, Iss: 2, pp 697-725
TL;DR: In this article, a powerful and flexible MCMC algorithm for stochastic simulation is introduced, based on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139-1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method.
Abstract: We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139-1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.

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Citations
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Journal ArticleDOI
TL;DR: It is shown here how it is possible to build efficient high dimensional proposal distributions by using sequential Monte Carlo methods, which allows not only to improve over standard Markov chain Monte Carlo schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so.
Abstract: Summary. Markov chain Monte Carlo and sequential Monte Carlo methods have emerged as the two main tools to sample from high dimensional probability distributions. Although asymptotic convergence of Markov chain Monte Carlo algorithms is ensured under weak assumptions, the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and/or if highly correlated variables are updated independently. We show here how it is possible to build efficient high dimensional proposal distributions by using sequential Monte Carlo methods. This allows us not only to improve over standard Markov chain Monte Carlo schemes but also to make Bayesian inference feasible for a large class of statistical models where this was not previously so. We demonstrate these algorithms on a non-linear state space model and a Levy-driven stochastic volatility model.

1,869 citations

Journal ArticleDOI
TL;DR: A flexible and robust simulation-based framework to infer demographic parameters from the site frequency spectrum (SFS) computed on large genomic datasets and shows that it allows one to study evolutionary models of arbitrary complexity, which cannot be tackled by other current likelihood-based methods.
Abstract: We introduce a flexible and robust simulation-based framework to infer demographic parameters from the site frequency spectrum (SFS) computed on large genomic datasets. We show that our composite-likelihood approach allows one to study evolutionary models of arbitrary complexity, which cannot be tackled by other current likelihood-based methods. For simple scenarios, our approach compares favorably in terms of accuracy and speed with , the current reference in the field, while showing better convergence properties for complex models. We first apply our methodology to non-coding genomic SNP data from four human populations. To infer their demographic history, we compare neutral evolutionary models of increasing complexity, including unsampled populations. We further show the versatility of our framework by extending it to the inference of demographic parameters from SNP chips with known ascertainment, such as that recently released by Affymetrix to study human origins. Whereas previous ways of handling ascertained SNPs were either restricted to a single population or only allowed the inference of divergence time between a pair of populations, our framework can correctly infer parameters of more complex models including the divergence of several populations, bottlenecks and migration. We apply this approach to the reconstruction of African demography using two distinct ascertained human SNP panels studied under two evolutionary models. The two SNP panels lead to globally very similar estimates and confidence intervals, and suggest an ancient divergence (>110 Ky) between Yoruba and San populations. Our methodology appears well suited to the study of complex scenarios from large genomic data sets.

1,199 citations

Journal ArticleDOI
TL;DR: Although the method arose in population genetics, ABC is increasingly used in other fields, including epidemiology, systems biology, ecology, and agent-based modeling, and many of these applications are briefly described.
Abstract: In the past 10 years a statistical technique, approximate Bayesian computation (ABC), has been developed that can be used to infer parameters and choose between models in the complicated scenarios that are often considered in the environmental sciences. For example, based on gene sequence and microsatellite data, the method has been used to choose between competing models of human demographic history as well as to infer growth rates, times of divergence, and other parameters. The method fits naturally in the Bayesian inferential framework, and a brief overview is given of the key concepts. Three main approaches to ABC have been developed, and these are described and compared. Although the method arose in population genetics, ABC is increasingly used in other fields, including epidemiology, systems biology, ecology, and agent-based modeling, and many of these applications are briefly described.

981 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a 3D map of the entire sky, covering three-quarters of the sky out to a distance of several kiloparsecs, based on Pan-STARRS 1 (PS1) and 2MASS photometry.
Abstract: We present a three-dimensional map of interstellar dust reddening, covering three-quarters of the sky out to a distance of several kiloparsecs, based on Pan-STARRS 1 (PS1) and 2MASS photometry. The map reveals a wealth of detailed structure, from filaments to large cloud complexes. The map has a hybrid angular resolution, with most of the map at an angular resolution of 3.4¢ –13.7¢ , and a maximum distance resolution of ~25%. The three-dimensional distribution of dust is determined in a fully probabilistic framework, yielding the uncertainty in the reddening distribution along each line of sight, as well as stellar distances, reddenings, and classifications for 800 million stars detected by PS1. We demonstrate the consistency of our reddening estimates with those of two-dimensional emission-based maps of dust reddening. In particular, we find agreement with the Planck t353GHz -based reddening map to within 0.05 mag in E ( ) B V - to a depth of 0.5 mag, and explore systematics at reddenings less than E ( ) B V - » 0.08 mag. We validate our per-star reddening estimates by comparison with reddening estimates for stars with both Sloan Digital Sky Survey photometry and Sloan Extension for Galactic Understanding and Exploration spectral classifications, finding per-star agreement to within 0.1 mag out to a stellar E ( ) B V - of 1 mag. We compare our map to two existing three-dimensional dust maps, by Marshall et al. and Lallement et al., demonstrating our finer angular resolution, and better distance resolution compared to the former within ~3 kpc. The map can be queried or downloaded at http://argonaut.skymaps.info. We expect the three-dimensional reddening map presented here to find a wide range of uses, among them correcting for reddening and extinction for objects embedded in the plane of the Galaxy, studies of Galactic structure, calibration of future emission-based dust maps, and determining distances to objects of known reddening.

536 citations

Journal ArticleDOI
TL;DR: This work shows how to construct appropriate summary statistics for ABC in a semi‐automatic manner, and shows that optimal summary statistics are the posterior means of the parameters.
Abstract: Summary. Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference.

527 citations

References
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Book
01 Jan 1993
TL;DR: This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.
Abstract: Meyn & Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

5,931 citations

Journal ArticleDOI
TL;DR: It is proved that Rao-Blackwellization causes a one-lag delay for the autocovariances among dependent samples obtained from data augmentation, and consequently, the mixture approximation produces estimates with smaller variances than the empirical approximation.
Abstract: SUMMARY We study the covariance structure of a Markov chain generated by the Gibbs sampler, with emphasis on data augmentation. When applied to a Bayesian missing data problem, the Gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector: the empirical distribution based on the sampled values of the parameter vector, and a mixture of complete data posteriors. We prove that Rao-Blackwellization causes a one-lag delay for the autocovariances among dependent samples obtained from data augmentation, and consequently, the mixture approximation produces estimates with smaller variances than the empirical approximation. The covariance structure results are used to compare different augmentation schemes. It is shown that collapsing and grouping random components in a Gibbs sampler with two or three components usually result in more efficient sampling schemes.

606 citations

Journal ArticleDOI
01 Jul 2003-Genetics
TL;DR: A new general method is introduced that samples independent genealogical histories using importance sampling (IS) and then samples other parameters with Markov chain Monte Carlo (MCMC) and it is concluded that these have an approximately equivalent effect.
Abstract: This article introduces a new general method for genealogical inference that samples independent genealogical histories using importance sampling (IS) and then samples other parameters with Markov chain Monte Carlo (MCMC). It is then possible to more easily utilize the advantages of importance sampling in a fully Bayesian framework. The method is applied to the problem of estimating recent changes in effective population size from temporally spaced gene frequency data. The method gives the posterior distribution of effective population size at the time of the oldest sample and at the time of the most recent sample, assuming a model of exponential growth or decline during the interval. The effect of changes in number of alleles, number of loci, and sample size on the accuracy of the method is described using test simulations, and it is concluded that these have an approximately equivalent effect. The method is used on three example data sets and problems in interpreting the posterior densities are highlighted and discussed.

492 citations

Journal ArticleDOI
TL;DR: In this article, the geometric ergodicity of Markov chains has been studied for multidimensional Hastings and Metropolis algorithms, and sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution π are given.
Abstract: We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and Metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution π. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the Metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given ; these show that the negative-definiteness property is not redundant.

465 citations

Journal ArticleDOI
TL;DR: Exact computable rates of convergence for Gaussian target distributions are obtained and different random and non‐random updating strategies and blocking combinations are compared using the rates.
Abstract: In this paper many convergence issues concerning the implementation of the Gibbs sampler are investigated. Exact computable rates of convergence for Gaussian target distributions are obtained. Different random and non-random updating strategies and blocking combinations are compared using the rates. The effect of dimensionality and correlation structure on the convergence rates are studied. Some examples are considered to demonstrate the results. For a Gaussian image analysis problem several updating strategies are described and compared. For problems in Bayesian linear models several possible parameterizations are analysed in terms of their convergence rates characterizing the optimal choice.

448 citations