# 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

**10.5** Problems

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

(10.1.1)

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 ...