Direct pivotal predictive inference

Abstract

Without assuming any pdf for some measured parameter, we derive a predictive pdf for the outcome of a second measurement, given the outcome of the first measurement and two common assumptions about the noise. These are that (1) it is additive, and (2) it is of some known pdf. The argument is based on a Bayesian analysis of the noise when no pdf is provided for the value of the parameter. In this way we avoid assuming an ad-hoc prior. We clarify how this method of direct predictive inference is distinct from fiducial prediction. We specify the distinct flaw in the fiducial argument, and outline the importance of this development in the foundations of probability and statistics. Keywords: nonparametric predictive inference, direct pivotal argument, pivotal argument, fiducial argument, fiducial prediction, Bayesian inference, reference prior, reference class.