An Axiomatic Framework for Belief Updates

TL;DR

This paper establishes an axiomatic framework for belief updates, demonstrating that such updates in probabilistic contexts are essentially monotonic transformations of likelihood ratios, thus formalizing the concept within Bayesian reasoning.

Contribution

It introduces axioms for belief updates and proves they imply belief updates are monotonic transformations of likelihood ratios, formalizing the belief update process.

Findings

Belief updates must satisfy specific axioms.

Belief updates are equivalent to monotonic transformations of likelihood ratios.

Provides a formal foundation for belief change in Bayesian systems.

Abstract

In the 1940's, a physicist named Cox provided the first formal justification for the axioms of probability based on the subjective or Bayesian interpretation. He showed that if a measure of belief satisfies several fundamental properties, then the measure must be some monotonic transformation of a probability. In this paper, measures of change in belief or belief updates are examined. In the spirit of Cox, properties for a measure of change in belief are enumerated. It is shown that if a measure satisfies these properties, it must satisfy other restrictive conditions. For example, it is shown that belief updates in a probabilistic context must be equal to some monotonic transformation of a likelihood ratio. It is hoped that this formal explication of the belief update paradigm will facilitate critical discussion and useful extensions of the approach.