Monte Carlo methods in Bayesian computation
type: Item
instance of: scholarly article
title: Monte Carlo methods in Bayesian computation
zbMATH Open document ID: 0949.65005
DOI: 10.1007/978-1-4612-1276-8
author: Ming-Hui Chen
author: Joseph G. Ibrahim
author: Q. M. Shao
published in: Springer Series in Statistics
publication date: 2000-03-15T00:00:00Z
review text: Bayesian computation (BC) means computation of interesting quantities from a posterior distribution as its normalizing constant, moments etc. Only in rare cases, e.g. if one has a conjugate prior, it is easy to perform it. In most cases, however, there are problems of higher computational complexity. Common, Basic Monte Carlo (BMC) methods may be rather inaccurate tools for solving numerical problems. In their further development, special advanced techniques have been found as Gibbs sampling, the Metropolis-Hastings algorithm etc. All these are joined under the notion Markov chain Monte Carlo (MCMC). MCMC is the tool for BC, and under BC, the authors understand sampling from a posterior distribution by MCMC. The book contains 10 chapters. Chapter 1 gives an outline in what follows and four interesting practical examples, the treatment of which goes through the book as a red thread. Chapter 2 gives the reader an excellent survey on MCMC with many varieties and propositions on their improvement and convergence. In Chapter 3 it is shown how BMC can be used for parameter estimation of a posterior distribution. Chapter 4 deals with estimation of posterior densities of interesting parameters (here emphasized as marginal densities), e.g. by kernel estimators. The determination of the normalizing constant of a posterior distribution is a special problem. To it, Chapter 5 is dedicated. The next Chapters 6 to 9 deal with constrained parameter problems, the calculation of credible intervals and intervals of highest probability density, comparing of nonnested models and variable selection, resp. Further interesting topics are summarized in the final Chapter 10. The authors form an excellent team of specialists in the given field. They write in a mathematically exact style, without superfluous sentences. A special advantage of the text is the permanent reference to the work of other authors and to previous chapters and formulas of the book what makes reading easy. The proofs of …
Mathematics Subject Classification ID: 65C60
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62-01
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 1414620
zbMATH Keywords: Markov chain Monte Carlo method
zbMATH Keywords: Bayesian parameter estimation
zbMATH Keywords: textbook
zbMATH Keywords: computational complexity
zbMATH Keywords: basic Monte Carlo methods
zbMATH Keywords: convergence
zbMATH Keywords: exercises
describes a project that uses: tsbridge
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4612-1276-8
OpenAlex ID: W2024315245
description: scientific article; zbMATH DE number 1414620