A-g Reference Informative Prior: A Note on Zellner's g Prior

Abstract

We consider a normal regression model and propose a natural conjugate reference informa- tive prior for the parameters, which takes into account different degrees of certainty about the different independent variables. In many applied problems, it is very important to assess a prior density function for the parameters of the model which is as informative as possible of the real opinion of the experts. A correct assessment of the prior density in fact can sometimes support insufficiency of data. In case of a normal Multiple Regression Model (MRM) y=Xfl+u unN(O, 2I) many different procedures for assessing the prior density for ,B and o2 in the natural conjugate form have been developed, but for many of those models it is difficult to evaluate the prior covariances for the elements of 8 (Zellner, 1983). For such situations and others, Zellner (1983) proposed a procedure for assessing a g-Reference Informative Prior (g-RIP) essentially based on the following: (a) before observing y, a conceptual sample yo is assumed generated by a model a2 yo=Xfl+ uo #o-N(O, -In) g with g>O, given. (b) values fl,a and o2 for ,B and o2 anticipated by the experts are assessed. (c) Muth's rational expectation hypothesis is invoked. The Zellner's g-RIP is effective in the analysis of engineering systems (Calvi et al., 1986) or in modelling biological phenomena. In these cases, subjective knowledge can often be formalised in the assignment of parameter values fla and oa2, as anticipated by the experts, and in the indication of a degree of precision g of the conceptual sample Yo. However further information is sometimes available about the roles of the indivi- dual independent variables in the model. In order to take into account such informa- tion, we propose an extension of Zellner's approach.