An adaptive population importance sampler

Monte Carlo (MC) methods are widely used in signal processing, machine learning and communications for statistical inference and stochastic optimization. A well-known class of MC methods is composed of importance sampling and its adaptive extensions (e.g., population Monte Carlo). In this work, we introduce an adaptive importance sampler using a population of proposal densities. The novel algorithm provides a global estimation of the variables of interest iteratively, using all the samples generated. The cloud of proposals is adapted by learning from a subset of previously generated samples, in such a way that local features of the target density can be better taken into account compared to single global adaptation procedures. Numerical results show the advantages of the proposed sampling scheme in terms of mean absolute error and robustness to initialization

Efficient Sequential Monte-Carlo Samplers for Bayesian Inference

In many problems, complex non-Gaussian and/or nonlinear models are required to accurately describe a physical system of interest. In such cases, Monte Carlo algorithms are remarkably flexible and extremely powerful approaches to solve such inference problems. However, in the presence of a high-dimensional and/or multimodal posterior distribution, it is widely documented that standard Monte-Carlo techniques could lead to poor performance. In this paper, the study is focused on a Sequential Monte-Carlo (SMC) sampler framework, a more robust and efficient Monte Carlo algorithm. Although this approach presents many advantages over traditional Monte-Carlo methods, the potential of this emergent technique is however largely underexploited in signal processing. In this work, we aim at proposing some novel strategies that will improve the efficiency and facilitate practical implementation of the SMC sampler specifically for signal processing applications. Firstly, we propose an automatic and adaptive strategy that selects the sequence of distributions within the SMC sampler that minimizes the asymptotic variance of the estimator of the posterior normalization constant. This is critical for performing model selection in modelling applications in Bayesian signal processing. The second original contribution we present improves the global efficiency of the SMC sampler by introducing a novel correction mechanism that allows the use of the particles generated through all the iterations of the algorithm (instead of only particles from the last iteration). This is a significant contribution as it removes the need to discard a large portion of the samples obtained, as is standard in standard SMC methods. This will improve estimation performance in practical settings where computational budget is important to consider.Comment: arXiv admin note: text overlap with arXiv:1303.3123 by other author

Bayesian Methods for Analysis and Adaptive Scheduling of Exoplanet Observations

We describe work in progress by a collaboration of astronomers and statisticians developing a suite of Bayesian data analysis tools for extrasolar planet (exoplanet) detection, planetary orbit estimation, and adaptive scheduling of observations. Our work addresses analysis of stellar reflex motion data, where a planet is detected by observing the "wobble" of its host star as it responds to the gravitational tug of the orbiting planet. Newtonian mechanics specifies an analytical model for the resulting time series, but it is strongly nonlinear, yielding complex, multimodal likelihood functions; it is even more complex when multiple planets are present. The parameter spaces range in size from few-dimensional to dozens of dimensions, depending on the number of planets in the system, and the type of motion measured (line-of-sight velocity, or position on the sky). Since orbits are periodic, Bayesian generalizations of periodogram methods facilitate the analysis. This relies on the model being linearly separable, enabling partial analytical marginalization, reducing the dimension of the parameter space. Subsequent analysis uses adaptive Markov chain Monte Carlo methods and adaptive importance sampling to perform the integrals required for both inference (planet detection and orbit measurement), and information-maximizing sequential design (for adaptive scheduling of observations). We present an overview of our current techniques and highlight directions being explored by ongoing research.Comment: 29 pages, 11 figures. An abridged version is accepted for publication in Statistical Methodology for a special issue on astrostatistics, with selected (refereed) papers presented at the Astronomical Data Analysis Conference (ADA VI) held in Monastir, Tunisia, in May 2010. Update corrects equation (3

Unbiased and Consistent Nested Sampling via Sequential Monte Carlo

We introduce a new class of sequential Monte Carlo methods called Nested Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested Sampling method of Skilling (2006) in terms of sequential Monte Carlo techniques. This new framework allows convergence results to be obtained in the setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An additional benefit is that marginal likelihood estimates are unbiased. In contrast to NS, the analysis of NS-SMC does not require the (unrealistic) assumption that the simulated samples be independent. As the original NS algorithm is a special case of NS-SMC, this provides insights as to why NS seems to produce accurate estimates despite a typical violation of its assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels in an automated manner via a preliminary pilot run, and present a new method for appropriately choosing the number of MCMC repeats at each iteration. Finally, a numerical study is conducted where the performance of NS-SMC and temperature-annealed SMC is compared on several challenging and realistic problems. MATLAB code for our experiments is made available at https://github.com/LeahPrice/SMC-NS .Comment: 45 pages, some minor typographical errors fixed since last versio

Sequential Monte Carlo with transformations

This paper examines methodology for performing Bayesian inference sequentially on a sequence of posteriors on spaces of different dimensions. For this, we use sequential Monte Carlo samplers, introducing the innovation of using deterministic transformations to move particles effectively between target distributions with different dimensions. This approach, combined with adaptive methods, yields an extremely flexible and general algorithm for Bayesian model comparison that is suitable for use in applications where the acceptance rate in reversible jump Markov chain Monte Carlo is low. We use this approach on model comparison for mixture models, and for inferring coalescent trees sequentially, as data arrives

An Adaptive Interacting Wang-Landau Algorithm for Automatic Density Exploration

While statisticians are well-accustomed to performing exploratory analysis in the modeling stage of an analysis, the notion of conducting preliminary general-purpose exploratory analysis in the Monte Carlo stage (or more generally, the model-fitting stage) of an analysis is an area which we feel deserves much further attention. Towards this aim, this paper proposes a general-purpose algorithm for automatic density exploration. The proposed exploration algorithm combines and expands upon components from various adaptive Markov chain Monte Carlo methods, with the Wang-Landau algorithm at its heart. Additionally, the algorithm is run on interacting parallel chains -- a feature which both decreases computational cost as well as stabilizes the algorithm, improving its ability to explore the density. Performance is studied in several applications. Through a Bayesian variable selection example, the authors demonstrate the convergence gains obtained with interacting chains. The ability of the algorithm's adaptive proposal to induce mode-jumping is illustrated through a trimodal density and a Bayesian mixture modeling application. Lastly, through a 2D Ising model, the authors demonstrate the ability of the algorithm to overcome the high correlations encountered in spatial models.Comment: 33 pages, 20 figures (the supplementary materials are included as appendices

On computational tools for Bayesian data analysis

While Robert and Rousseau (2010) addressed the foundational aspects of Bayesian analysis, the current chapter details its practical aspects through a review of the computational methods available for approximating Bayesian procedures. Recent innovations like Monte Carlo Markov chain, sequential Monte Carlo methods and more recently Approximate Bayesian Computation techniques have considerably increased the potential for Bayesian applications and they have also opened new avenues for Bayesian inference, first and foremost Bayesian model choice.Comment: This is a chapter for the book "Bayesian Methods and Expert Elicitation" edited by Klaus Bocker, 23 pages, 9 figure

Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo (AdPMCMC)

We develop a novel advanced Particle Markov chain Monte Carlo algorithm that is capable of sampling from the posterior distribution of non-linear state space models for both the unobserved latent states and the unknown model parameters. We apply this novel methodology to five population growth models, including models with strong and weak Allee effects, and test if it can efficiently sample from the complex likelihood surface that is often associated with these models. Utilising real and also synthetically generated data sets we examine the extent to which observation noise and process error may frustrate efforts to choose between these models. Our novel algorithm involves an Adaptive Metropolis proposal combined with an SIR Particle MCMC algorithm (AdPMCMC). We show that the AdPMCMC algorithm samples complex, high-dimensional spaces efficiently, and is therefore superior to standard Gibbs or Metropolis Hastings algorithms that are known to converge very slowly when applied to the non-linear state space ecological models considered in this paper. Additionally, we show how the AdPMCMC algorithm can be used to recursively estimate the Bayesian Cram\'er-Rao Lower Bound of Tichavsk\'y (1998). We derive expressions for these Cram\'er-Rao Bounds and estimate them for the models considered. Our results demonstrate a number of important features of common population growth models, most notably their multi-modal posterior surfaces and dependence between the static and dynamic parameters. We conclude by sampling from the posterior distribution of each of the models, and use Bayes factors to highlight how observation noise significantly diminishes our ability to select among some of the models, particularly those that are designed to reproduce an Allee effect

Bayesian computational methods

In this chapter, we will first present the most standard computational challenges met in Bayesian Statistics, focussing primarily on mixture estimation and on model choice issues, and then relate these problems with computational solutions. Of course, this chapter is only a terse introduction to the problems and solutions related to Bayesian computations. For more complete references, see Robert and Casella (2004, 2009), or Marin and Robert (2007), among others. We also restrain from providing an introduction to Bayesian Statistics per se and for comprehensive coverage, address the reader to Robert (2007), (again) among others.Comment: This is a revised version of a chapter written for the Handbook of Computational Statistics, edited by J. Gentle, W. Hardle and Y. Mori in 2003, in preparation for the second editio