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 = (y1, …, yn)′ 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, 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
where the elements of β and ...