Statistical Intervals, 2nd Edition

Appendix H Basic Results from Bayesian Inference Models

Introduction

This appendix describes and provides background information and results used mainly in the construction of Bayesian intervals in Chapters 15–17. The appendix contains:

• Some useful technical results that are used to obtain the posterior distributions and posterior predictive distributions in this appendix (Section H.1).
• A formal presentation of Bayes’ theorem (Section H.2).
• The definition of conjugate prior distributions with examples for the binomial, Poisson, and normal distributions (Section H.3).
• The definition of Jeffreys prior distributions with examples for the binomial, Poisson, and normal distributions. Also a modified Jeffreys prior distribution for the normal distribution (Section H.4).
• The definition of posterior predictive distributions with examples, using conjugate prior distributions, for the binomial, Poisson, and normal distributions (Section H.5).
• Posterior predictive distributions for the binomial and the Poisson distributions using Jeffreys prior distributions. Also a posterior predictive distribution for the normal distributions using a modified Jeffreys prior distribution (Section H.6).

H.1 Basic Results Used in This Appendix

In this appendix we use the following results:

1. We frequently need to compute the marginal distribution corresponding to a mixture of two normal distributions. The following result is useful in making those computations.

   Consider the continuous normal-normal ...