Posterior Bayes factor analysis for an exponential regression model

In the exponential regression model, Bayesian inference concerning the non-linear regression parameter has proved extremely difficult. In particular, standard improper diffuse priors for the usual parameters lead to an improper posterior for the non-linear regression parameter. In a recent paper Ye and Berger (1991) applied the reference prior approach of Bernardo (1979) and Berger and Bernardo (1989) yielding a proper informative prior for . This prior depends on the values of the explanatory variable, goes to 0 as goes to 1, and depends on the specification of a hierarchical ordering of importance of the parameters.This paper explains the failure of the uniform prior to give a proper posterior: the reason is the appearance of the determinant of the information matrix in the posterior density for . We apply the posterior Bayes factor approach of Aitkin (1991) to this problem; in this approach we integrate out nuisance parameters with respect to their conditional posterior density given the parameter of interest. The resulting integrated likelihood for requires only the standard diffuse prior for all the parameters, and is unaffected by orderings of importance of the parameters. Computation of the likelihood for is extremely simple. The approach is applied to the three examples discussed by Berger and Ye and the likelihoods compared with their posterior densities.