Learning Convex Sets of Probability from Data

Abstract

Several theories of inference and decision employ sets of probability distributions as the fundamental representation of (subjective) belief. This paper investigates a frequentist connection between empirical data and convex sets of probability distributions. Building on earlier work by Walley and Fine, a framework is advanced in which a sequence of random outcomes can be described as being drawn from a convex set of distributions, rather than just from a single distribution. The extra generality can be detected from observable characteristics of the outcome sequence. The paper presents new asymptotic convegence results paralleling the laws of large numbers in probability theory, and concludes with a comparison between this approach and approaches based on prior subjective constraints.