# Chapter 16 Bayesian Statistical Intervals for the Binomial, Poisson, and Normal Distributions

## Objectives and Overview

This chapter describes the construction of Bayesian intervals for data generated from the distributions discussed in Chapters 3 and 4 (normal distribution), Chapter 6 (binomial distribution) and Chapter 7 (Poisson distribution). We extend these methods (for the same distributions) in Chapter 17 to consider the construction of Bayesian intervals for the more complicated situation involving hierarchical models.

The topics discussed in this chapter are:

• The construction of Bayesian intervals for the binomial distribution (Section 16.1).
• The construction of Bayesian intervals for the Poisson distribution (Section 16.2).
• The construction of Bayesian intervals for the normal distribution (Section 16.3).

In each section we show how to compute credible intervals for the distribution parameter(s), functions of the parameter(s), and Bayesian tolerance and prediction intervals.

As we saw in Chapter 15, Bayesian methods using appropriately chosen diffuse prior distributions result in credible intervals that are close to the confidence intervals obtained using non-Bayesian methods. For particular examples, in Chapters 6 and 7 we used non-Bayesian methods to construct confidence intervals for the parameters of the binomial and Poisson distributions, respectively (or functions thereof). One of these methods—and one which we recommended for practical use—was the Jeffreys ...

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