Aitkin M. Statistical Inference: An Integrated Bayesian/Likelihood Approach

Chapman & Hall/CRC, 2010. — 238 p. — (Monographs on Statistics & Applied Probability 116). — ISBN: 1420093436, 978-1420093438.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
Example, Statistical models, The likelihood function, Theories:
Pure likelihood theory,
Bayesian theory,
Likelihood-based repeated sampling theory,
Model-guided survey sampling theory;
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