Fiducial Prediction and Semi-Bayesian Inference

Abstract

We investigate the problem of fiducial prediction for unobserved quantities within the framework of the functional model described previously by Dawid and Stone. It is supposed that these are related to a completely unknown parameter by means of a regular functional model, and that the observations are either given as known functions of the predictands, or are themselves related to them by means of a functional model. We develop algebraic conditions which allow the application of fiducial logic to the prediction problem, and explore the consequences of such an application--some of which appear unacceptable unless still stronger conditions are imposed. A reinterpretation of the fiducial prediction problem is given which can be applied to yield an inferential distribution for the unknown parameter in the presence of partial prior information, expressible as a functional hypermodel for the parameter, governed by a completely unknown hyperparameter. This solution agrees with the fiducial distribution when the hypermodel is vacuous and with the Bayes posterior distribution when the hyperparameter is fully known, but allows in addition for intermediate levels of partial prior knowledge.