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) equation

where y = (y1, …, yn)′ is an (n × 1) vector of observations, X = (x1, …, xn)′ 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, σ2In, γo) (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 σ2 is a diffuse (noninformative/vague/flat) prior with the pdf

(10.1.2) equation

where the elements of β and ...

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