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Books on the topic 'Gaussian; Markov chain Monte Carlo methods'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 February 2022

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1

Liang, Faming, Chuanhai Liu, and Raymond J. Carroll. Advanced Markov Chain Monte Carlo Methods. Chichester, UK: John Wiley & Sons, Ltd, 2010. http://dx.doi.org/10.1002/9780470669723.

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2

Joseph, Anosh. Markov Chain Monte Carlo Methods in Quantum Field Theories. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-46044-0.

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3

Liang, F. Advanced Markov chain Monte Carlo methods: Learning from past samples. Hoboken, NJ: Wiley, 2010.

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4

Winkler, Gerhard. Image Analysis, Random Fields and Markov Chain Monte Carlo Methods. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/978-3-642-55760-6.

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5

Neal, Radford M. Markov chain Monte Carlo methods based on "slicing" the density function. Toronto: University of Toronto, Dept. of Statistics, 1997.

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6

Gerhard, Winkler. Image analysis, random fields and Markov chain Monte Carlo methods: A mathematical introduction. 2nd ed. Berlin: Springer, 2003.

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7

1946-, Winkler Gerhard, ed. Image analysis, random fields and Markov chain Monte Carlo methods: A mathematical introduction. 2nd ed. Berlin: Springer, 2003.

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8

Cheng, Russell. Finite Mixture Examples; MAPIS Details. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198505044.003.0018.

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Two detailed numerical examples are given in this chapter illustrating and comparing mainly the reversible jump Markov chain Monte Carlo (RJMCMC) and the maximum a posteriori/importance sampling (MAPIS) methods. The numerical examples are the well-known galaxy data set with sample size 82, and the Hidalgo stamp issues thickness data with sample size 485. A comparison is made of the estimates obtained by the RJMCMC and MAPIS methods for (i) the posterior k-distribution of the number of components, k, (ii) the predictive finite mixture distribution itself, and (iii) the posterior distributions of the component parameters and weights. The estimates obtained by MAPIS are shown to be more satisfactory and meaningful. Details are given of the practical implementation of MAPIS for five non-normal mixture models, namely: the extreme value, gamma, inverse Gaussian, lognormal, and Weibull. Mathematical details are also given of the acceptance-rejection importance sampling used in MAPIS.

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9

Carroll, Raymond, Faming Liang, and Chuanhai Liu. Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley & Sons, Incorporated, John, 2011.

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10

Carroll, Raymond, Faming Liang, and Chuanhai Liu. Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley & Sons, Incorporated, John, 2010.

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11

Carroll, Raymond, Faming Liang, and Chuanhai Liu. Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley & Sons, Incorporated, John, 2011.

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12

Allen, Michael P., and Dominic J. Tildesley. Monte Carlo methods. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803195.003.0004.

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The estimation of integrals by Monte Carlo sampling is introduced through a simple example. The chapter then explains importance sampling, and the use of the Metropolis and Barker forms of the transition matrix defined in terms of the underlying matrix of the Markov chain. The creation of an appropriately weighted set of states in the canonical ensemble is described in detail and the method is extended to the isothermal–isobaric, grand canonical and semi-grand ensembles. The Monte Carlo simulation of molecular fluids and fluids containing flexible molecules using a reptation algorithm is discussed. The parallel tempering or replica exchange method for more efficient exploration of the phase space is introduced, and recent advances including solute tempering and convective replica exchange algorithms are described.

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13

Boudreau, Joseph F., and Eric S. Swanson. Monte Carlo methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198708636.003.0007.

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Monte Carlo methods are those designed to obtain numerical answers with the use of random numbers . This chapter discusses random engines, which provide a pseudo-random pattern of bits, and their use in for sampling a variety of nonuniform distributions, for both continuous and discrete variables. A wide selection of uniform and nonuniform variate generators from the C++ standard library are reviewed, and common techniques for generating custom nonuniform variates are discussed. The chapter presents the uses of Monte Carlo to evaluate integrals, particularly multidimensional integrals, and then introduces the important method of Markov chain Monte Carlo, suitable for solving a wide range of scientific problems that require the sampling of complicated multivariate distributions. Relevant topics in probability and statistics are also introduced in this chapter. Finally, the topics of thermalization, autocorrelation, multimodality, and Gibbs sampling are presented.

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14

Joseph, Anosh. Markov Chain Monte Carlo Methods in Quantum Field Theories: A Modern Primer. Springer, 2020.

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15

Winkler, Gerhard. Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction (Stochastic Modelling and Applied Probability). 2nd ed. Springer, 2006.

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16

Martin, Andrew D. Bayesian Analysis. Edited by Janet M. Box-Steffensmeier, Henry E. Brady, and David Collier. Oxford University Press, 2009. http://dx.doi.org/10.1093/oxfordhb/9780199286546.003.0021.

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This article surveys modern Bayesian methods of estimating statistical models. It first provides an introduction to the Bayesian approach for statistical inference, contrasting it with more conventional approaches. It then explains the Monte Carlo principle and reviews commonly used Markov Chain Monte Carlo (MCMC) methods. This is followed by a practical justification for the use of Bayesian methods in the social sciences, and a number of examples from the literature where Bayesian methods have proven useful are shown. The article finally provides a review of modern software for Bayesian inference, and a discussion of the future of Bayesian methods in political science. One area ripe for research is the use of prior information in statistical analyses. Mixture models and those with discrete parameters (such as change point models in the time-series context) are completely underutilized in political science.

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17

Geweke, John, Gary Koop, and Herman Van Dijk, eds. The Oxford Handbook of Bayesian Econometrics. Oxford University Press, 2011. http://dx.doi.org/10.1093/oxfordhb/9780199559084.001.0001.

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Bayesian econometric methods have enjoyed an increase in popularity in recent years. Econometricians, empirical economists, and policymakers are increasingly making use of Bayesian methods. The Oxford Handbook of Bayesian Econometrics is a single source about Bayesian methods in specialized fields. It contains articles by leading Bayesians on the latest developments in their specific fields of expertise. The volume provides broad coverage of the application of Bayesian econometrics in the major fields of economics and related disciplines, including macroeconomics, microeconomics, finance, and marketing. It reviews the state of the art in Bayesian econometric methodology, with articles on posterior simulation and Markov chain Monte Carlo methods, Bayesian nonparametric techniques, and the specialized tools used by Bayesian time series econometricians such as state space models and particle filtering. It also includes articles on Bayesian principles and methodology.

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18

Henderson, Daniel A., R. J. Boys, Carole J. Proctor, and Darren J. Wilkinson. Linking systems biology models to data: A stochastic kinetic model of p53 oscillations. Edited by Anthony O'Hagan and Mike West. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198703174.013.7.

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This article discusses the use of a stochastic kinetic model to study protein level oscillations in single living cancer cells, using the p53 and Mdm2 proteins as examples. It describes the refinement of a dynamic stochastic process model of the cellular response to DNA damage and compares this model to time course data on the levels of p53 and Mdm2. The article first provides a biological background on p53 and Mdm2 before explaining how the stochastic kinetic model is constructed. It then introduces the stochastic kinetic model and links it to the data and goes on to apply sophisticated MCMC methods to compute posterior distributions. The results demonstrate that it is possible to develop computationally intensive Markov chain Monte Carlo (MCMC) methods for conducting a Bayesian analysis of an intra-cellular stochastic systems biology model using single-cell time course data.

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19

Cheng, Russell. Finite Mixture Models. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198505044.003.0017.

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Fitting a finite mixture model when the number of components, k, is unknown can be carried out using the maximum likelihood (ML) method though it is non-standard. Two well-known Bayesian Markov chain Monte Carlo (MCMC) methods are reviewed and compared with ML: the reversible jump method and one using an approximating Dirichlet process. Another Bayesian method, to be called MAPIS, is examined that first obtains point estimates for the component parameters by the maximum a posteriori method for different k and then estimates posterior distributions, including that for k, using importance sampling. MAPIS is compared with ML and the MCMC methods. The MCMC methods produce multimodal posterior parameter distributions in overfitted models. This results in the posterior distribution of k being biased towards high k. It is shown that MAPIS does not suffer from this problem. A simple numerical example is discussed.

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20

Coolen, A. C. C., A. Annibale, and E. S. Roberts. Graphs with hard constraints: further applications and extensions. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198709893.003.0007.

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This chapter looks at further topics pertaining to the effective use of Markov Chain Monte Carlo to sample from hard- and soft-constrained exponential random graph models. The chapter considers the question of how moves can be sampled efficiently without introducing unintended bias. It is shown mathematically and numerically that apparently very similar methods of picking out moves can give rise to significant differences in the average topology of the networks generated by the MCMC process. The general discussion in complemented with pseudocode in the relevant section of the Algorithms chapter, which explicitly sets out some accurate and practical move sampling approaches. The chapter also describes how the MCMC equilibrium probabilities can be purposely deformed to, for example, target desired correlations between degrees of connected nodes. The mathematical exposition is complemented with graphs showing the results of numerical simulations.

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21

Lopes, Hedibert, and Nicholas Polson. Analysis of economic data with multiscale spatio-temporal models. Edited by Anthony O'Hagan and Mike West. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198703174.013.12.

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This article discusses the use of Bayesian multiscale spatio-temporal models for the analysis of economic data. It demonstrates the utility of a general modelling approach for multiscale analysis of spatio-temporal processes with areal data observations in an economic study of agricultural production in the Brazilian state of Espìrito Santo during the period 1990–2005. The article first describes multiscale factorizations for spatial processes before presenting an exploratory multiscale data analysis and explaining the motivation for multiscale spatio-temporal models. It then examines the temporal evolution of the underlying latent multiscale coefficients and goes on to introduce a Bayesian analysis based on the multiscale decomposition of the likelihood function along with Markov chain Monte Carlo (MCMC) methods. The results from agricultural production analysis show that the spatio-temporal framework can effectively analyse massive economics data sets.

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22

Quintana, José Mario, Carlos Carvalho, James Scott, and Thomas Costigliola. Extracting S&P500 and NASDAQ Volatility: The Credit Crisis of 2007–2008. Edited by Anthony O'Hagan and Mike West. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198703174.013.13.

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This article demonstrates the utility of Bayesian modelling and inference in financial market volatility analysis, using the 2007-2008 credit crisis as a case study. It first describes the applied problem and goal of the Bayesian analysis before introducing the sequential estimation models. It then discusses the simulation-based methodology for inference, including Markov chain Monte Carlo (MCMC) and particle filtering methods for filtering and parameter learning. In the study, Bayesian sequential model choice techniques are used to estimate volatility and volatility dynamics for daily data for the year 2007 for three market indices: the Standard and Poor’s S&P500, the NASDAQ NDX100 and the financial equity index called XLF. Three models of financial time series are estimated: a model with stochastic volatility, a model with stochastic volatility that also incorporates jumps in volatility, and a Garch model.

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23

Rubin, Donald, Xiaoqin Wang, Li Yin, and Elizabeth Zell. Bayesian causal inference: Approaches to estimating the effect of treating hospital type on cancer survival in Sweden using principal stratification. Edited by Anthony O'Hagan and Mike West. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198703174.013.24.

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This article discusses the use of Bayesian causal inference, and more specifically the posterior predictive approach of Rubin’s causal model (RCM) and methods of principal stratification, in estimating the effects of ‘treating hospital type’ on cancer survival. Using the Karolinska Institute in Stockholm, Sweden, as a case study, the article investigates which type of hospital (large patient volume vs. small volume) is superior for treating certain serious conditions. The study examines which factors may reasonably be considered ignorable in the context of covariates available, as well as non-compliance complications due to transfers between hospital types for treatment. The article first provides an overview of the general Bayesian approach to causal inference, primarily with ignorable treatment assignment, before introducing the proposed approach and motivating it using simple method-of-moments summary statistics. Finally, the results of simulation using Markov chain Monte Carlo (MCMC) methods are presented.

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24

Laver, Michael, and Ernest Sergenti. Systematically Interrogating Agent-Based Models. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691139036.003.0004.

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This chapter develops the methods for designing, executing, and analyzing large suites of computer simulations that generate stable and replicable results. It starts with a discussion of the different methods of experimental design, such as grid sweeping and Monte Carlo parameterization. Next, it demonstrates how to calculate mean estimates of output variables of interest. It does so by first discussing stochastic processes, Markov Chain representations, and model burn-in. It focuses on three stochastic process representations: nonergodic deterministic processes that converge on a single state; nondeterministic stochastic processes for which a time average provides a representative estimate of the output variables; and nondeterministic stochastic processes for which a time average does not provide a representative estimate of the output variables. The estimation strategy employed depends on which stochastic process the simulation follows. Lastly, the chapter presents a set of diagnostic checks used to establish an appropriate sample size for the estimation of the means.

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