Ignorance and the Expressiveness of Single- and Set-Valued Probability Models of Belief

TL;DR

This paper explores how belief models using multiple probability sets can better represent human ignorance and partial beliefs than single distributions, especially with multiple exclusive options.

Contribution

It demonstrates that certain ignorance and belief patterns cannot be captured by a single set of distributions but can be represented by multiple sets, introducing a partial qualitative probability.

Findings

Single distributions cannot represent ignorance over three or more options.

Multiple sets of distributions can model partially ordered beliefs.

Partial qualitative probability shares attributes with standard probability distributions.

Abstract

Over time, there have hen refinements in the way that probability distributions are used for representing beliefs. Models which rely on single probability distributions depict a complete ordering among the propositions of interest, yet human beliefs are sometimes not completely ordered. Non-singleton sets of probability distributions can represent partially ordered beliefs. Convex sets are particularly convenient and expressive, but it is known that there are reasonable patterns of belief whose faithful representation require less restrictive sets. The present paper shows that prior ignorance about three or more exclusive alternatives and the emergence of partially ordered beliefs when evidence is obtained defy representation by any single set of distributions, but yield to a representation baud on several uts. The partial order is shown to be a partial qualitative probability which shares some intuitively appealing attributes with probability distributions.