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.
Theories of Statistical Inference
The likelihood function
Nonmodel-based repeated sampling
The Integrated Bayes/Likelihood Approach
The importance of parametrization
The simple/simple hypothesis testing problem
The simple/composite hypothesis testing problem
Posterior likelihood approach
The comparison of unrelated models
Example—GHQ score and psychiatric diagnosis
t-Tests and Normal Variance Tests
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
More general regression models
The multinomial model for multiple populations
Complex sample designs
A complex example
Regression and Analysis of Variance
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
The data augmentation algorithm
Two-level variance component models
Test for a zero variance component
Murray Aitkin is an honorary professorial fellow in the Department of Mathematics and Statistics at the University of Melbourne in Australia.
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