Abstract: While studying various features of the posterior distribution of a vector-valued parameter using an MCMC sample, systematically subsampling of the MCMC output can only lead to poorer estimation. Nevertheless, a 1-in-k subsample is often all that is retained in investigations where intensive computations are involved or where speed is essential. The goal of benchmark estimation is to produce a number of estimates based on the best available information, i.e. the entire MCMC sample, and to use these to improve other estimates made on the basis of the subsample. We take a simple approach by creating a weighted subsample where the weights are quickly obtained as a solution to a system of linear equations. We provide a theoretical basis for the method and illustrate the technique using examples from the literature. For a subsampling rate of 1-in-10, the observed reductions in MSE often exceed 50% for a number of posterior features. Much larger gains are expected for certain complex estimation methods and for the commonly used thinner subsampling rates. Benchmark estimation can be used wherever other fast or efficient estimation strategies like importance sampling already exist. We show how the two strategies can be used in conjunction with each other. We discuss some asymptotic properties of benchmark estimators that provide insight into the gains associated with the technique. The observed gains are found to closely match the theoretical values predicted by the asymptotic, even for k as small as 10.
Estimating rare event probabilities is a commonly encountered important problem in several engineering and scientific applications, most often observed in the form of probability of failure (PF) estimation or, alternatively and better sounding for the public, reliability estimation. In many practical applications, such as for structures, airplanes, mechanical equipment, and many more, failure probabilities are fortunately very low, from 10-4 to even 10-9 and less. Such estimations are of utmost importance for design choices, emergency preparedness, safety regulations, maintenance suggestions and more. Calculating such small numbers with accuracy however presents many numerical and mathematical challenges. To make matters worse, these estimations in realistic applications are usually based on high dimensional random spaces with numerous random variables and processes involved. A single simulation of such a model, or else a single model call, may also require several minutes to hours of computing time. As such, reducing the number of model calls is of great importance in these problems and one of the critical parameters that limits or prohibits use of several available techniques in the literature. This research is motivated by efficiently and precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on a developed framework termed Approximate Sampling Target with Postprocessing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. Hamiltonian Markov Chain Monte Carlo sampling is characterized by much better scalability, faster mixing rates, is capable of generating samples with much weaker auto-correlation, even in complex high-dimensional parameter spaces, and has enjoyed broad-spectrum successes in most general settings. HMCMC adopts physical system dynamics, rather than a proposal probability distribution, and can be used to produce distant proposal samples for the integrated Metropolis step, thereby avoiding the slow exploration of the state space that results from the diffusive behavior of simple random-walk proposals. In this work, we aim to advance knowledge on Hamiltonian Markov Chain Monte Carlo methods, in general, with particular emphasis on its efficient utilization for rare event probability estimation in both Gaussian and Non-Gaussian spaces. This research also seeks to offer significant advancements in probabilistic inference and reliability predictions. Thus, in this context, we develop various Quasi-Newton based HMCMC schemes, which can sample very adeptly, particularly in difficult cases of high curvature, high-dimensionality and very small failure probabilities. The methodology is formally introduced, and the key theoretical aspects, and the underlying assumptions are discussed. Performance of the proposed methodology is then compared against state-of-the-art Subset Simulation in a series of challenging static and dynamic (time-dependent reliability) low- and high-dimensional benchmark problems. In the last phase of this work, with an aim to avoid using analytical gradients, within the proposed HMCMC-based framework, we investigate application of the Automatic Differentiation (AD) technique. In addition, to avoid use of gradients altogether and to improve the performance of the original SuS algorithm, we study the application of Quasi-Newton based HMCMC within the Subset Simulation framework. Various numerical examples are then presented to showcase the performance of the aforementioned approaches.
Markov Chain Monte Carlo (MCMC) methods are now an indispensable tool in scientific computing. This book discusses recent developments of MCMC methods with an emphasis on those making use of past sample information during simulations. The application examples are drawn from diverse fields such as bioinformatics, machine learning, social science, combinatorial optimization, and computational physics. Key Features: Expanded coverage of the stochastic approximation Monte Carlo and dynamic weighting algorithms that are essentially immune to local trap problems. A detailed discussion of the Monte Carlo Metropolis-Hastings algorithm that can be used for sampling from distributions with intractable normalizing constants. Up-to-date accounts of recent developments of the Gibbs sampler. Comprehensive overviews of the population-based MCMC algorithms and the MCMC algorithms with adaptive proposals. This book can be used as a textbook or a reference book for a one-semester graduate course in statistics, computational biology, engineering, and computer sciences. Applied or theoretical researchers will also find this book beneficial.
While there have been few theoretical contributions on the Markov Chain Monte Carlo (MCMC) methods in the past decade, current understanding and application of MCMC to the solution of inference problems has increased by leaps and bounds. Incorporating changes in theory and highlighting new applications, Markov Chain Monte Carlo: Stochastic Simul
This volume contains the proceedings of the Workshop on Monte Carlo Methods held at The Fields Institute for Research in Mathematical Sciences (Toronto, 1998). The workshop brought together researchers in physics, statistics, and probability. The papers in this volume - of the invited speakers and contributors to the poster session - represent the interdisciplinary emphasis of the conference. Monte Carlo methods have been used intensively in many branches of scientific inquiry. Markov chain methods have been at the forefront of much of this work, serving as the basis of many numerical studies in statistical physics and related areas since the Metropolis algorithm was introduced in 1953. Statisticians and theoretical computer scientists have used these methods in recent years, working on different fundamental research questions, yet using similar Monte Carlo methodology. This volume focuses on Monte Carlo methods that appear to have wide applicability and emphasizes new methods, practical applications and theoretical analysis. It will be of interest to researchers and graduate students who study and/or use Monte Carlo methods in areas of probability, statistics, theoretical physics, or computer science.
This textbook explains the fundamentals of Markov Chain Monte Carlo (MCMC) without assuming advanced knowledge of mathematics and programming. MCMC is a powerful technique that can be used to integrate complicated functions or to handle complicated probability distributions. MCMC is frequently used in diverse fields where statistical methods are important – e.g. Bayesian statistics, quantum physics, machine learning, computer science, computational biology, and mathematical economics. This book aims to equip readers with a sound understanding of MCMC and enable them to write simulation codes by themselves. The content consists of six chapters. Following Chap. 2, which introduces readers to the Monte Carlo algorithm and highlights the advantages of MCMC, Chap. 3 presents the general aspects of MCMC. Chap. 4 illustrates the essence of MCMC through the simple example of the Metropolis algorithm. In turn, Chap. 5 explains the HMC algorithm, Gibbs sampling algorithm and Metropolis-Hastings algorithm, discussing their pros, cons and pitfalls. Lastly, Chap. 6 presents several applications of MCMC. Including a wealth of examples and exercises with solutions, as well as sample codes and further math topics in the Appendix, this book offers a valuable asset for students and beginners in various fields.
In a family study of breast cancer, epidemiologists in Southern California increase the power for detecting a gene-environment interaction. In Gambia, a study helps a vaccination program reduce the incidence of Hepatitis B carriage. Archaeologists in Austria place a Bronze Age site in its true temporal location on the calendar scale. And in France, researchers map a rare disease with relatively little variation. Each of these studies applied Markov chain Monte Carlo methods to produce more accurate and inclusive results. General state-space Markov chain theory has seen several developments that have made it both more accessible and more powerful to the general statistician. Markov Chain Monte Carlo in Practice introduces MCMC methods and their applications, providing some theoretical background as well. The authors are researchers who have made key contributions in the recent development of MCMC methodology and its application. Considering the broad audience, the editors emphasize practice rather than theory, keeping the technical content to a minimum. The examples range from the simplest application, Gibbs sampling, to more complex applications. The first chapter contains enough information to allow the reader to start applying MCMC in a basic way. The following chapters cover main issues, important concepts and results, techniques for implementing MCMC, improving its performance, assessing model adequacy, choosing between models, and applications and their domains. Markov Chain Monte Carlo in Practice is a thorough, clear introduction to the methodology and applications of this simple idea with enormous potential. It shows the importance of MCMC in real applications, such as archaeology, astronomy, biostatistics, genetics, epidemiology, and image analysis, and provides an excellent base for MCMC to be applied to other fields as well.
Markov chain Monte Carlo (e. g., the Metropolis algorithm and Gibbs sampler) is a general tool for simulation of complex stochastic processes useful in many types of statistical inference. The basics of Markov chain Monte Carlo are reviewed, including choice of algorithms and variance estimation, and some new methods are introduced. The use of Markov chain Monte Carlo for maximum likelihood estimation is explained, and its performance is compared with maximum pseudo likelihood estimation. Markov chain, Monte Carlo, Maximum likelihood, Metropolis algorithm, Gibbs sampler, Variance estimation.
