An Axiomatic Framework for Belief Updates
This work provides a foundational framework for belief updates, which is incremental as it extends Cox's justification for probability axioms to belief changes, potentially aiding in critical discussion and extensions in fields like statistics and AI.
The paper tackles the problem of formalizing measures of belief updates by proposing an axiomatic framework, showing that such measures must satisfy restrictive conditions like being monotonic transformations of likelihood ratios in a probabilistic context.
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.