A result in renyi's conditional probability theory with application to subjective probability

Abstract

Whenever we carry out a concrete sampling procedure to estimate the probability of an event, we are in fact estimating a conditional probability. Conditionalization is taken with respect to the population being sampled and with respect to the appropriateness of the sampling procedures (e.g., independence, randomness). The question arises, under what circumstances can we tit such conditional probabilities together so as to generate a unique absolute probability on a larger sample space containing the original spaces? This paper presents a development of Renyi’s theory of a conditional probability space (hereafter c.P.s.). Renyi’s theory is reviewed in Section 1. It emerges that this notion of a c.p.s. is not suitable for addressing the question raised above. In Section 2 the notion of a linked set of measurable spaces is introduced, and Section 3 gives the necessary and sufficient conditions that a set of probability spaces generate a unique c.p.s. in Renyi’s sense, given that the underlying measurable spaces are linked. Further results throw light on the question whether a c.p.s. generated in this fashion can be described by a normalizable measure. A final section is devoted to exploring briefly the question of whether non-normalizable belief states can arise in subjective probability. Received opinion holds that they cannot. I argue that this conclusion is premature, and rests on an unsatisfactory analysis of measurements for partial belief.