Decision Making for Symbolic Probability

TL;DR

This paper develops a decision theory for symbolic probability (SP), a framework that generalizes traditional probability by using symbolic supports, enabling preference modeling and uncertainty elicitation without numerical probabilities.

Contribution

It introduces a decision theory for SP with a utility function based on support pairs, extending subjective interpretation and decision-making to symbolic uncertainty.

Findings

Preference relations are represented by utility functions over support pairs.

Subjective interpretation of SP supports facilitates uncertainty elicitation.

The theory explains decision-making based on support comparison in symbolic probability.

Abstract

This paper proposes a decision theory for a symbolic generalization of probability theory (SP). Darwiche and Ginsberg [2,3] proposed SP to relax the requirement of using numbers for uncertainty while preserving desirable patterns of Bayesian reasoning. SP represents uncertainty by symbolic supports that are ordered partially rather than completely as in the case of standard probability. We show that a preference relation on acts that satisfies a number of intuitive postulates is represented by a utility function whose domain is a set of pairs of supports. We argue that a subjective interpretation is as useful and appropriate for SP as it is for numerical probability. It is useful because the subjective interpretation provides a basis for uncertainty elicitation. It is appropriate because we can provide a decision theory that explains how preference on acts is based on support comparison.