On conditional probability and bayesian inference

Abstract

Measurement theory has dealt with the applicability of the conditional probability formula to the updating of probability assignments when new information is incorporated. In this paper the original probability measure is taken as given, and an assumption on the relation between this probability and a possible conditional probability is imposed. Provided that the original probability is non-atomic, it is proved that there is one and only one transformed probability measure satisfying the assumption. Building on this result, we discuss the hypotheses underlying Bayesian inference. In the Bayesian parametric model, a joint probability distribution on the product of the sample space and the parameter space is assigned. As this probability distribution is shown to be non-atomic, we conclude that, apart from measure-theoretic representability hypotheses, the existence of this joint probability is the only nontechnical hypothesis underlying Bayesian parametric statistical inference.