Direct Bayes for Interest Parameters

For an interest parameterψ(θ) the Bayesian method eliminates the nuisance parameterλ(θ) by integrating with respect to a conditional prior for λ given ψ. This conditional prior may be difficult to specify in both the subjective and objective Bayesian contexts. We propose the use of results from likelihood theory that give highly accurate third order determinations of various marginal distributions in the continuous case. The appropriate conditional prior exists as part of the calculations for the marginal distributions and corresponds to an integration used in the marginalization. To third order however the integrated likelihood for the interest parameter can be obtained directly; it needs no model information beyond that commonly available for accurate analyses of likelihood. Simple examples are given where the steps are analytically available to illustrate this direct Bayesian calculation.