Aspects of the Predictive and Estimative Approaches in the Determination of Probabilities

Abstract

SUMMARY An exposition of the predictive (observabilistic) and estimative approaches for determining probabilities within a Bayesian framework is presented. It is demonstrated that the usual measurement error model can be subsumed under either the estimative or observabilistic modes and that a much more frequently occurring model should be treated in the latter mode. Because of mathematical tractability the simple exponential distribution is used to exhibit these concepts and a particular data set used to exemplify the type of calculations that are necessary and the inferences that can be made. In a previous paper (Geisser, 1971), I argued that the predictive approach in statistics which had long been neglected or entirely disregarded was a more appropriate vehicle for the transtnission of a statistical inference, practically speaking, than the classical estimative approach. The predictive approach couches inferences and decisions in terms of observables or potential observables or interesting functions thereof, while the estimative approach is involved with parameters. In this paper I shall attempt to present compelling arguments that predictivism or observabilism is also a generally superior conceptual framework for inference which can subsume the estimative approach, when valid, as a limiting case, particularly in the determination of probabilities. Predictive or observabilistic inference is directed towards statements about a finite number of observables that conceptually have or had the potential of being generated. Clearly this includes sample surveys or any kind of sampling from a finite population, since here involves statements about some function of the finite totality of observations from the population which is often misdesignated as a parameter'7. Passage to the limit will then include estimation since will be entities that arise from the prediction of a function of an infinite number of potential observables. Another situation where some difficulty in drawing a distinction between parameters and observables frequently arises, is in the measuring of some physical constant speed of light, length of a table, etc. with an imperfect instrument. Here the model is that the observed is the parameter or true value plus measurement error, say