Contents
1.1 Probability and Bayes’ Theorem
1.2 Examples on Bayes’ Theorem
Chapter 2: Bayesian inference for the normal distribution
2.1 Nature of Bayesian inference
2.2 Normal prior and likelihood
2.3 Several normal observations with a normal prior
2.8 HDRs for the normal variance
2.10 Conjugate prior distributions
2.12 Normal mean and variance both unknown
2.13 Conjugate joint prior for the normal distribution
Chapter 3: Some other common distributions
3.2 Reference prior for the binomial likelihood
3.6 Reference prior for the uniform distribution
3.8 The first digit problem; invariant priors
3.9 The circular normal distribution
3.10 Approximations based on the likelihood
3.11 Reference posterior distributions
4.2 One-sided hypothesis tests
4.4 Point (or sharp) null hypotheses with prior information
4.5 Point null hypotheses for the normal distribution
Chapter 5: Two-sample problems
5.1 Two-sample problems – both variances unknown
5.2 Variances unknown but equal
5.3 Variances unknown and unequal (Behrens–Fisher problem)
5.4 The Behrens–Fisher controversy
5.5 Inferences concerning a variance ratio
5.6 Comparison of two proportions; the $2 imes 2$ table
Chapter 6: Correlation, regression and the analysis of variance
6.1 Theory of the correlation coefficient
6.2 Examples on the use of the correlation coefficient
6.3 Regression and the bivariate normal model
6.4 Conjugate prior for the bivariate regression model
6.5 Comparison of several means – the one way model
7.2 The stopping rule principle
7.3 Informative stopping rules
7.4 The likelihood principle and reference priors
7.7 Decision theory and hypothesis testing
Chapter 8: Hierarchical models
8.1 The idea of a hierarchical model
8.2 The hierarchical normal model
8.5 Bayesian analysis for an unknown overall mean
8.6 The general linear model revisited
Chapter 9: The Gibbs sampler and other numerical methods
9.1 Introduction to numerical methods
9.3 Data augmentation by Monte Carlo
9.6 The Metropolis–Hastings algorithm
9.7 Introduction to WinBUGS and OpenBUGS
Chapter 10: Some approximate methods
10.1 Bayesian importance sampling
10.2 Variational Bayesian methods: simple case
10.3 Variational Bayesian methods: general case
10.4 ABC: Approximate Bayesian Computation
10.5 Reversible jump Markov chain Monte Carlo
Appendix A: Common statistical distributions
A.3 Normal approximation to chi-squared
A.5 Inverse chi-squared distribution
A.7 Log chi-squared distribution
A.9 Normal/chi-squared distribution
A.13 Negative binomial distribution
A.14 Hypergeometric distribution
A.17 Circular normal distribution
A.19 Snedecor’s F distribution
A.22 The probability that one beta variable is greater than another
A.23 Bivariate normal distribution
A.24 Multivariate normal distribution
A.25 Distribution of the correlation coefficient