Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also gives guidance on the choice of sample size and allocation <s> BIB001 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced into significance testing, general Bayesian inference and maximum likelihood estimation. This paper presents basic methodology of MCMC, emphasizing the Bayesian paradigm, conditional probability and the intimate relationship with Markov random fields in spatial statistics. Hastings algorithms are discussed, including Gibbs, Metropolis and some other variations. Pairwise difference priors are described and are used subsequently in three Bayesian applications, in each of which there is a pronounced spatial or temporal aspect to the modeling. The examples involve logistic regression in the presence of unobserved covariates and ordinal factors; the analysis of agricultural field experiments, with adjustment for fertility gradients; and processing of low-resolution medical images obtained by a gamma camera. Additional methodological issues arise in each of these applications and in the Appendices. The paper lays particular emphasis on the calculation of posterior probabilities and concurs with others in its view that MCMC facilitates a fundamental breakthrough in applied Bayesian modeling. <s> BIB002 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter. <s> BIB003 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> INTRODUCING MARKOV CHAIN MONTE CARLO Introduction The Problem Markov Chain Monte Carlo Implementation Discussion HEPATITIS B: A CASE STUDY IN MCMC METHODS Introduction Hepatitis B Immunization Modelling Fitting a Model Using Gibbs Sampling Model Elaboration Conclusion MARKOV CHAIN CONCEPTS RELATED TO SAMPLING ALGORITHMS Markov Chains Rates of Convergence Estimation The Gibbs Sampler and Metropolis-Hastings Algorithm INTRODUCTION TO GENERAL STATE-SPACE MARKOV CHAIN THEORY Introduction Notation and Definitions Irreducibility, Recurrence, and Convergence Harris Recurrence Mixing Rates and Central Limit Theorems Regeneration Discussion FULL CONDITIONAL DISTRIBUTIONS Introduction Deriving Full Conditional Distributions Sampling from Full Conditional Distributions Discussion STRATEGIES FOR IMPROVING MCMC Introduction Reparameterization Random and Adaptive Direction Sampling Modifying the Stationary Distribution Methods Based on Continuous-Time Processes Discussion IMPLEMENTING MCMC Introduction Determining the Number of Iterations Software and Implementation Output Analysis Generic Metropolis Algorithms Discussion INFERENCE AND MONITORING CONVERGENCE Difficulties in Inference from Markov Chain Simulation The Risk of Undiagnosed Slow Convergence Multiple Sequences and Overdispersed Starting Points Monitoring Convergence Using Simulation Output Output Analysis for Inference Output Analysis for Improving Efficiency MODEL DETERMINATION USING SAMPLING-BASED METHODS Introduction Classical Approaches The Bayesian Perspective and the Bayes Factor Alternative Predictive Distributions How to Use Predictive Distributions Computational Issues An Example Discussion HYPOTHESIS TESTING AND MODEL SELECTION Introduction Uses of Bayes Factors Marginal Likelihood Estimation by Importance Sampling Marginal Likelihood Estimation Using Maximum Likelihood Application: How Many Components in a Mixture? Discussion Appendix: S-PLUS Code for the Laplace-Metropolis Estimator MODEL CHECKING AND MODEL IMPROVEMENT Introduction Model Checking Using Posterior Predictive Simulation Model Improvement via Expansion Example: Hierarchical Mixture Modelling of Reaction Times STOCHASTIC SEARCH VARIABLE SELECTION Introduction A Hierarchical Bayesian Model for Variable Selection Searching the Posterior by Gibbs Sampling Extensions Constructing Stock Portfolios With SSVS Discussion BAYESIAN MODEL COMPARISON VIA JUMP DIFFUSIONS Introduction Model Choice Jump-Diffusion Sampling Mixture Deconvolution Object Recognition Variable Selection Change-Point Identification Conclusions ESTIMATION AND OPTIMIZATION OF FUNCTIONS Non-Bayesian Applications of MCMC Monte Carlo Optimization Monte Carlo Likelihood Analysis Normalizing-Constant Families Missing Data Decision Theory Which Sampling Distribution? Importance Sampling Discussion STOCHASTIC EM: METHOD AND APPLICATION Introduction The EM Algorithm The Stochastic EM Algorithm Examples GENERALIZED LINEAR MIXED MODELS Introduction Generalized Linear Models (GLMs) Bayesian Estimation of GLMs Gibbs Sampling for GLMs Generalized Linear Mixed Models (GLMMs) Specification of Random-Effect Distributions Hyperpriors and the Estimation of Hyperparameters Some Examples Discussion HIERARCHICAL LONGITUDINAL MODELLING Introduction Clinical Background Model Detail and MCMC Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC FOR NONLINEAR HIERARCHICAL MODELS Introduction Implementing MCMC Comparison of Strategies A Case Study from Pharmacokinetics-Pharmacodynamics Extensions and Discussion BAYESIAN MAPPING OF DISEASE Introduction Hypotheses and Notation Maximum Likelihood Estimation of Relative Risks Hierarchical Bayesian Model of Relative Risks Empirical Bayes Estimation of Relative Risks Fully Bayesian Estimation of Relative Risks Discussion MCMC IN IMAGE ANALYSIS Introduction The Relevance of MCMC to Image Analysis Image Models at Different Levels Methodological Innovations in MCMC Stimulated by Imaging Discussion MEASUREMENT ERROR Introduction Conditional-Independence Modelling Illustrative examples Discussion GIBBS SAMPLING METHODS IN GENETICS Introduction Standard Methods in Genetics Gibbs Sampling Approaches MCMC Maximum Likelihood Application to a Family Study of Breast Cancer Conclusions MIXTURES OF DISTRIBUTIONS: INFERENCE AND ESTIMATION Introduction The Missing Data Structure Gibbs Sampling Implementation Convergence of the Algorithm Testing for Mixtures Infinite Mixtures and Other Extensions AN ARCHAEOLOGICAL EXAMPLE: RADIOCARBON DATING Introduction Background to Radiocarbon Dating Archaeological Problems and Questions Illustrative Examples Discussion Index <s> BIB004 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> The Markov chain Monte Carlo (MCMC) method, as a computer-intensive statistical tool, has enjoyed an enormous upsurge in interest over the last few years. This paper provides a simple, comprehensive and tutorial review of some of the most common areas of research in this field. We begin by discussing how MCMC algorithms can be constructed from standard building-blocks to produce Markov chains with the desired stationary distribution. We also motivate and discuss more complex ideas that have been proposed in the literature, such as continuous time and dimension jumping methods. We discuss some implementational issues associated with MCMC methods. We take a look at the arguments for and against multiple replications, consider how long chains should be run for and how to determine suitable starting points. We also take a look at graphical models and how graphical approaches can be used to simplify MCMC implementation. Finally, we present a couple of examples, which we use as case-studies to highlight some of the points made earlier in the text. In particular, we use a simple changepoint model to illustrate how to tackle a typical Bayesian modelling problem via the MCMC method, before using mixture model problems to provide illustrations of good sampler output and of the implementation of a reversible jump MCMC algorithm. <s> BIB005 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> Abstract This article describes a new Metropolis-like transition rule, the multiple-try Metropolis, for Markov chain Monte Carlo (MCMC) simulations. By using this transition rule together with adaptive direction sampling, we propose a novel method for incorporating local optimization steps into a MCMC sampler in continuous state-space. Numerical studies show that the new method performs significantly better than the traditional Metropolis-Hastings (M-H) sampler. With minor tailoring in using the rule, the multiple-try method can also be exploited to achieve the effect of a griddy Gibbs sampler without having to bear with griddy approximations, and the effect of a hit-and-run algorithm without having to figure out the required conditional distribution in a random direction. <s> BIB006 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> This paperback edition is a reprint of the 2001 Springer edition. This book provides a self-contained and up-to-date treatment of the Monte Carlo method and develops a common framework under which various Monte Carlo techniques can be "standardized" and compared. Given the interdisciplinary nature of the topics and a moderate prerequisite for the reader, this book should be of interest to a broad audience of quantitative researchers such as computational biologists, computer scientists, econometricians, engineers, probabilists, and statisticians. It can also be used as the textbook for a graduate-level course on Monte Carlo methods. Many problems discussed in the alter chapters can be potential thesis topics for masters or Ph.D. students in statistics or computer science departments. Jun Liu is Professor of Statistics at Harvard University, with a courtesy Professor appointment at Harvard Biostatistics Department. Professor Liu was the recipient of the 2002 COPSS Presidents' Award, the most prestigious one for statisticians and given annually by five leading statistical associations to one individual under age 40. He was selected as a Terman Fellow by Stanford University in 1995, as a Medallion Lecturer by the Institute of Mathematical Statistics (IMS) in 2002, and as a Bernoulli Lecturer by the International Bernoulli Society in 2004. He was elected to the IMS Fellow in 2004 and Fellow of the American Statistical Association in 2005. He and co-workers have published more than 130 research articles and book chapters on Bayesian modeling and computation, bioinformatics, genetics, signal processing, stochastic dynamic systems, Monte Carlo methods, and theoretical statistics. "An excellent survey of current Monte Carlo methods. The applications amply demonstrate the relevance of this approach to modern computing. The book is highly recommended." (Mathematical Reviews) "This book provides comprehensive coverage of Monte Carlo methods, and in the process uncovers and discusses commonalities among seemingly disparate techniques that arose in various areas of application. The book is well organized; the flow of topics follows a logical development. The coverage is up-to-date and comprehensive, and so the book is a good resource for people conducting research on Monte Carlo methods. The book would be an excellent supplementary text for a course in scientific computing ." (SIAM Review) "The strength of this book is in bringing together advanced Monte Carlo (MC) methods developed in many disciplines. Throughout the book are examples of techniques invented, or reinvented, in different fields that may be applied elsewhere. Those interested in using MC to solve difficult problems will find many ideas, collected from a variety of disciplines, and references for further study." (Technometrics) <s> BIB007 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly, it discusses new interesting research horizons. <s> BIB008 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> Bayesian Theory.- Off-line Distributional Approximations and the Variational Bayes Method.- Principal Component Analysis and Matrix Decompositions.- Functional Analysis of Medical Image Sequences.- On-line Inference of Time-Invariant Parameters.- On-line Inference of Time-Variant Parameters.- The Mixture-based Extension of the AR Model (MEAR).- Concluding Remarks. <s> BIB009 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> The Multiple Try Metropolis (MTM) method is a generalization of the classical Metropolis-Hastings algorithm in which the next state of the chain is chosen among a set of samples, according to normalized weights. In the literature, several extensions have been proposed. In this work, we show and remark upon the flexibility of the design of MTM-type methods, fulfilling the detailed balance condition. We discuss several possibilities and show different numerical results. <s> BIB010 </s> Survey on Probabilistic Models of Low-Rank Matrix Factorizations <s> Gibbs Sampling <s> From the basics to the forefront of modern research, this book presents all aspects of probability theory, statistics and data analysis from a Bayesian perspective for physicists and engineers. The book presents the roots, applications and numerical implementation of probability theory, and covers advanced topics such as maximum entropy distributions, stochastic processes, parameter estimation, model selection, hypothesis testing and experimental design. In addition, it explores state-of-the art numerical techniques required to solve demanding real-world problems. The book is ideal for students and researchers in physical sciences and engineering. <s> BIB011
As a powerful framework for sampling from a probability distribution, Markov chain Monte Carlo (MCMC) methods, also called Markov chain simulation, construct a Markov chain such that its equilibrium distribution is the desired distribution  BIB001 BIB002 BIB004 BIB005  BIB007 BIB008 . Random walk Monte Carlo methods are a large subclass of MCMC and they include mainly Metropolis-Hastings [54, 62, BIB009 BIB011 , Gibbs sampling, Slice sampling [54, 62, BIB009 , Multiple-try Metropolis BIB006 BIB010 . Gibbs sampling or Gibbs sampler, a simple MCMC algorithm, is especially applicable for approximating a sequence of observations by a specified probability distribution when direct sampling is intractable. In addition, the basic version of Gibbs sampling is also a special case of the Metropolis-Hastings algorithm [54, 62] . In detailed implementation, Gibbs sampling adopts the strategy of sampling from a conditional probability distribution instead of marginalizing the joint probability distribution by integrating over other variables. In other words, Gibbs sampling generates alternatively an instance from its corresponding conditional probability distribution by fixing other variables. Let z 1 , z 2 , . . . , z M be M blocks of random variables and set z = (z 1 , z 2 , . . . , z M ). The joint probability distribution of z is written as p(z) = p(z 1 , z 2 , . . . , z M ). In this subsection, we only consider the case that it is difficult to directly sampling from the joint probability distribution p(z). To this end, we can sample each component z i from marginal distribution in order. The marginal distribution of z i can be theoretically obtained by the following formulation: Generally speaking, the integrating term in the above formula is not tractable. A wise sampling strategy is that we generate z i according to the conditional probability distribution p(z i |z\z i ), where . By using Bayes' rule, the relationship between the conditional probability distribution and the joint probability distribution is given as follows: The sampling procedure is repeated by cycling through all variables. We summarize the outline of Gibbs sampling as below. In the above sampling procedure, T is the maximum number of iterations. Therefore, we have T To alleviate the influence of random initializations, we can ignore the samples within the burn-in period. Although Gibbs sampling is commonly efficient in obtaining marginal distributions from conditional distributions, it can be highly inefficient for some cases such as sampling from Mexican hat like distribution BIB003 . The Metropolis algorithm is an instance of MCMC algorithms and walks randomly with an acceptance/rejection rule until the convergence is achieved. As the generalization of Metropolis algorithm, the Metropolis-Hastings algorithm modifies the jumping rules and its convergence speed is improved BIB009 . Broadly speaking, the advanced Gibbs sampling algorithm can be regarded as a special case of the Metropolis-Hastings algorithm. The Gibbs sampler is applicable for models that are conditionally conjugate, while the Metropolis algorithm can be used for not conditionally conjugate models. Hence, we can combine both the Gibbs sampler and the Metropolis algorithm to sample from complex distributions BIB009 .