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AuthorTanner, Martin A. author
TitleTools for Statistical Inference [electronic resource] : Methods for the Exploration of Posterior Distributions and Likelihood Functions / by Martin A. Tanner
ImprintNew York, NY : Springer US, 1993
Edition Second Edition
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This book provides a unified introduction to a variety of computational algorithms for likelihood and Bayesian inference. In this second edition, I have attempted to expand the treatment of many of the techniques disยญ cussed, as well as include important topics such as the Metropolis algorithm and methods for assessing the convergence of a Markov chain algorithm. Prerequisites for this book include an understanding of mathematical statistics at the level of Bickel and Doksum (1977), some understanding of the Bayesian approach as in Box and Tiao (1973), experience with condiยญ tional inference at the level of Cox and Snell (1989) and exposure to statistical models as found in McCullagh and Neider (1989). I have chosen not to present the proofs of convergence or rates of convergence since these proofs may require substantial background in Markov chain theory which is beyond the scope ofthis book. However, references to these proofs are given. There has been an explosion of papers in the area of Markov chain Monte Carlo in the last five years. I have attempted to identify key references - though due to the volatility of the field some work may have been missed


1 Introduction -- 2 Normal Approximations to Likelihoods and to Posteriors -- 2.1 Likelihood/Posterior Density -- 2.2 Maximum Likelihood -- 2.3 Normal-Based Inference -- 2.4 The ?-Method (Propagation of Errors) -- 2.5 Highest Posterior Density Regions -- 3 Nonnormal Approximations to Likelihoods and to Posteriors -- 3.1 Conjugate Priors and Numerical Integration -- 3.2 Posterior Moments and Marginalization Based on Laplaceโ{128}{153}s Method -- 3.3 Monte Carlo Methods -- 4 The EM Algorithm -- 4.1 Introduction -- 4.2 Theory -- 4.3 EM in the Exponential Family -- 4.4 Standard Errors in the Context of EM -- 5 The Data Augmentation Algorithm -- 5.1 Introduction and Motivation -- 5.2 Computing and Sampling from the Predictive Distribution -- 5.3 Calculating the Content and Boundary of the HPD Region -- 5.4 Remarks on the General Implementation of the Data Augmentation Algorithm -- 5.5 Overview of the Convergence Theory of Data Augmentation -- 5.6 Poor Manโ{128}{153}s Data Augmentation Algorithms -- 5.7 Sampling/Importance Resampling (SIR) -- 5.8 General Imputation Methods -- 5.9 Further Importance Sampling Ideas -- 5.10 Sampling in the Context of Multinomial Data -- 6 Markov Chain Monte Carlo: The Gibbs Sampler and the Metropolis Algorithm -- 6.1 Introduction to the Gibbs Sampler -- 6.2 Examples -- 6.3 Assessing Convergence of the Chain -- 6.4 The Griddy Gibbs Sampler -- 6.5 The Metropolis Algorithm -- 6.6 Conditional Inference via the Gibbs Sampler -- References

Mathematics Applied mathematics Engineering mathematics Mathematics Applications of Mathematics


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