In the exponential regression model, Bayesian inference concerning the non-linear regression parameter p 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 p. This prior depends on the values of the explanatory variable, goes to 0 as p 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 p. 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 p 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 p is extremely simple. The approach is applied to the three examples discussed by Berger and Ye and the likelihoods compared with their posterior densities.