# CHAPTER 10

# BAYESIAN ANALYSIS

## Outline

**10.1** Introduction (Zellner’s Model)

**10.2** Conditional Bayesian Inference

**10.3** Matrix Variate t-Distribution

**10.4** Bayesian Analysis in Multivariate Regression Model

Bayesian analysis has become an influential topic in modern statistics. In this chapter we discuss the Bayesian analysis when the error distribution is the multivariate t-model.

# 10.1 Introduction (Zellner’s Model)

Zellner (1976) was first to initiate the use of multivariate t-error in from a Bayesian analysis of regression models. In his seminal paper, he considered linear multivariate t-regression models under Bayesian viewpoint. We consider the multiple regression model (7.1.1) as

where ** y** = (

*y*

_{1}, …,

*y*

_{n})′ is an (

*n*× 1) vector of observations,

**= (**

*X*

*x*′_{1}, …,

*x*′_{n})′ is a nonstochastic (

*n*×

*p*) matrix of full rank

*p*,

**β**is a (

*p*× 1) vector of unknown regression coefficients, and ε is an (

*n*× 1) random error-vector distributed as

*M*

^{(n)}

_{t}(

**0**, σ

^{2}

*I*_{n}, γ

_{o}) (in this case). With this error assumption, we get into the theory for uncorrelated but dependent errors, making the analysis more applicable in real-life situations.

To begin with, we assume that the prior knowledge for **β** and σ^{2} is a diffuse (noninformative/vague/flat) prior with the pdf

(10.1.2)

where the elements of **β** and ...

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