Intuitions about Ordered Beliefs Leading to Probabilistic Models

TL;DR

This paper offers a new rationale for probabilistic belief models by deriving qualitative probability from simple assumptions about belief relationships, avoiding betting arguments and emphasizing ordered belief structures.

Contribution

It introduces a novel derivation of probability models from qualitative belief assumptions, expanding the theoretical foundation beyond traditional axiomatic justifications.

Findings

Qualitative probability can be derived from five basic assumptions about beliefs.

Additional assumptions bridge qualitative probability to full probability models.

The approach supports models based on complete and partial belief orderings.

Abstract

The general use of subjective probabilities to model belief has been justified using many axiomatic schemes. For example, ?consistent betting behavior' arguments are well-known. To those not already convinced of the unique fitness and generality of probability models, such justifications are often unconvincing. The present paper explores another rationale for probability models. ?Qualitative probability,' which is known to provide stringent constraints on belief representation schemes, is derived from five simple assumptions about relationships among beliefs. While counterparts of familiar rationality concepts such as transitivity, dominance, and consistency are used, the betting context is avoided. The gap between qualitative probability and probability proper can be bridged by any of several additional assumptions. The discussion here relies on results common in the recent AI literature, introducing a sixth simple assumption. The narrative emphasizes models based on unique complete orderings, but the rationale extends easily to motivate set-valued representations of partial orderings as well.