Updating Sets of Probabilities

TL;DR

This paper examines the justification for conditioning in the context of sets of probability measures, revealing that axioms supporting conditioning for single measures do not extend straightforwardly to sets, and explores alternative update methods.

Contribution

It provides an axiomatic analysis showing the limitations of van Fraassen's axioms for updating sets of measures and introduces additional axioms that justify conditioning in this broader context.

Findings

van Fraassen's axioms are insufficient for sets of measures

additional axioms are needed to justify conditioning

alternative update methods are possible when axioms are relaxed

Abstract

There are several well-known justifications for conditioning as the appropriate method for updating a single probability measure, given an observation. However, there is a significant body of work arguing for sets of probability measures, rather than single measures, as a more realistic model of uncertainty. Conditioning still makes sense in this context--we can simply condition each measure in the set individually, then combine the results--and, indeed, it seems to be the preferred updating procedure in the literature. But how justified is conditioning in this richer setting? Here we show, by considering an axiomatic account of conditioning given by van Fraassen, that the single-measure and sets-of-measures cases are very different. We show that van Fraassen's axiomatization for the former case is nowhere near sufficient for updating sets of measures. We give a considerably longer (and not as compelling) list of axioms that together force conditioning in this setting, and describe other update methods that are allowed once any of these axioms is dropped.