How to quantify the coherence of a set of beliefs

TL;DR

This paper introduces methods to quantify the conflict among probability estimates for events and to find a single distribution that best represents conflicting beliefs, with applications in language models and expert forecasts.

Contribution

It characterizes the structure of coherent probability estimates and applies these insights to belief elicitation and forecast merging.

Findings

Characterizes the polytope of coherent probability estimates.

Proposes methods to quantify belief conflict.

Demonstrates applications in language models and forecast merging.

Abstract

Given conflicting probability estimates for a set of events, how can we quantify how much they conflict? How can we find a single probability distribution that best encapsulates the given estimates? One approach is to minimize a loss function such as binary KL-divergence that quantifies the dissimilarity between the given estimates and the candidate probability distribution. Given a set of events, we characterize the facets of the polytope of coherent probability estimates about those events. We explore two applications of these ideas: eliciting the beliefs of large language models, and merging expert forecasts into a single coherent forecast.