Statistical Inference: An Integrated Bayesian/Likelihood Approach, 1st Edition (Hardback) book cover

Statistical Inference

An Integrated Bayesian/Likelihood Approach, 1st Edition

By Murray Aitkin

Chapman and Hall/CRC

254 pages | 95 B/W Illus.

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pub: 2010-06-02
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Description

Filling a gap in current Bayesian theory, Statistical Inference: An Integrated Bayesian/Likelihood Approach presents a unified Bayesian treatment of parameter inference and model comparisons that can be used with simple diffuse prior specifications. This novel approach provides new solutions to difficult model comparison problems and offers direct Bayesian counterparts of frequentist t-tests and other standard statistical methods for hypothesis testing.

After an overview of the competing theories of statistical inference, the book introduces the Bayes/likelihood approach used throughout. It presents Bayesian versions of one- and two-sample t-tests, along with the corresponding normal variance tests. The author then thoroughly discusses the use of the multinomial model and noninformative Dirichlet priors in "model-free" or nonparametric Bayesian survey analysis, before covering normal regression and analysis of variance. In the chapter on binomial and multinomial data, he gives alternatives, based on Bayesian analyses, to current frequentist nonparametric methods. The text concludes with new goodness-of-fit methods for assessing parametric models and a discussion of two-level variance component models and finite mixtures.

Emphasizing the principles of Bayesian inference and Bayesian model comparison, this book develops a unique methodology for solving challenging inference problems. It also includes a concise review of the various approaches to inference.

Reviews

This interesting book on model selection provides a nice review of the frequentist, likelihood, and Bayesian approaches to inference and model comparison. … an interesting book on model selection.

—Cavan Reilly, Journal of the American Statistical Association, December 2011

I like this book very much … a worthy new tool based on the posterior distribution of the likelihood with good examples of its capabilities and limitations. … I strongly recommend the book, enjoy Aitken’s writing style, and recognize his many strong contributions to the methodological Bayesian literature.

—Tom Burr, Technometrics, November 2011

This is a stimulating book that should be of interest to Bayesians and statisticians with a general interest in statistical inference. … It is likely to be controversial, even heretical, to Bayesians. However, this is precisely why it is worth reading: in exploring the new ideas, whether we ultimately accept them or not, we gain a better understanding of the current orthodoxy. … one of the interesting contributions of the book is the discussion of the use of Bayes factors — if not ‘from the inside,’ at least from someone who has been thinking deeply about them for some time. … The book contains some useful points that are known but ought to be better known, and it is useful to have a reference to them. … I am pleased to have had the opportunity to read it.

—A.H. Welsh, Australian & New Zealand Journal of Statistics, 2011

The emphasis on evidence rather than decision theory makes the book especially relevant to scientific investigations. It gives interesting and thoughtful comparisons to alternative approaches to inference … The very deep and solid inferential foundations the book lays support a carefully thought out and impressive superstructure, covering topics which include variance component models, finite mixtures, regression, anova, complex survey designs, and other topics. It would provide a valuable and thought-provoking volume for advanced students studying the foundations of inference and their practical implications. It would make a particularly good book for a reading group.

—David Hand, International Statistical Review (2011), 79, 1

Table of Contents

Theories of Statistical Inference

Example

Statistical models

The likelihood function

Theories

Nonmodel-based repeated sampling

Conclusion

The Integrated Bayes/Likelihood Approach

Introduction

Probability

Prior ignorance

The importance of parametrization

The simple/simple hypothesis testing problem

The simple/composite hypothesis testing problem

Posterior likelihood approach

Bayes factors

The comparison of unrelated models

Example—GHQ score and psychiatric diagnosis

t-Tests and Normal Variance Tests

One-sample t-test

Two samples: equal variances

The two-sample test

Two samples: different variances

The normal model variance

Variance heterogeneity test

Unified Analysis of Finite Populations

Sample selection indicators

The Bayesian bootstrap

Sampling without replacement

Regression models

More general regression models

The multinomial model for multiple populations

Complex sample designs

A complex example

Discussion

Regression and Analysis of Variance

Multiple regression

Nonnested models

Binomial and Multinomial Data

Single binomial samples

Single multinomial samples

Two-way tables for correlated proportions

Multiple binomial samples

Two-way tables for categorical responses—no fixed margins

Two-way tables for categorical responses—one fixed margin

Multinomial "nonparametric" analysis

Goodness of Fit and Model Diagnostics

Frequentist model diagnostics

Bayesian model diagnostics

The posterior predictive distribution

Multinomial deviance computation

Model comparison through posterior deviances

Examples

Simulation study

Discussion

Complex Models

The data augmentation algorithm

Two-level variance component models

Test for a zero variance component

Finite mixtures

References

Author Index

Subject Index

About the Author/Editors

Murray Aitkin is an honorary professorial fellow in the Department of Mathematics and Statistics at the University of Melbourne in Australia.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General