# Chapter 9Bayesian Inference

**Package(s):** `LearnBayes`

, `VGAM`

## 9.1 Introduction

We encountered the Bayes formula in Section 5.6 and some important Bayesian sampling distributions in Section 6.9. Bayesian probabilities are of interest in themselves and such probabilities also highlight the requirement for appropriate choice of prior distributions for unknown parameters. The Bayesian probabilities will be discussed in depth in Section 9.2. Useful theoretical aspects of the Bayesian paradigm are then discussed in Section 9.3. Bayesian inference for standard probability distributions will be taken up in Sections 9.4.1–9.4.5. An advantage of the Bayes paradigm is that a single approach helps to answer the three inferential problems of estimation, intervals, and testing hypotheses, and we will consider *credible intervals* in Section 9.5. Finally, hypotheses testing problem will be dealt with in Section 9.6 using *Bayes factors*.

## 9.2 Bayesian Probabilities

In general, almost all Bayesian texts assume that the reader learned probability theory in a separate course and so there is almost no discussion of *Bayesian probabilities*. As a matter of factual non-coincidence, an observation has been that either the word “Bayes” does not appear in many classical probability texts, or merely gets a simple honorable mention. This is not a surprise as most Bayesian texts define “posterior distribution” only in the context of data and not as a probability model. There are a lot of surprises when we consider ...