A Theory of Higher Order Probabilities

Abstract

We set up a general framework for higher order probabilities. A simple HOP (Higher Order Probability space) consists of a probability space and an operation PR, such that, for every event A and every real closed interval zL PR(A,A) is the event that A's "true " probability lies in A. (The 'true" probability can be construed here either as the objective probability, or the probability assigned by an expert, or the one assigned eventually in a fuller state of knowledge.) In a general HOP the operation PR has also an additional argument ranging over an ordered set of time-points, or, more generally, over a partially ordered set of stages; PR(A,t,z~) is the event that A's probability at stage t lies in A. First we investigate simple HOPs and then the general ones. Assuming some intuitively justified axioms, we derive the most general structure of such a space. We also indicate various connections with modal logic. IA part of this paper has been included in a talk given in a NSF symposium on foundations of probability and causality, organized by W. Harper and B. Skyrms at UC irvine, July 1985.! wish to thank the organizers for the opportunity to discuss and clarify some of these ideas.