Constriction for sets of probabilities

TL;DR

This paper explores how belief updating procedures can tighten probability bounds for events, demonstrating that constriction can occur both with and without evidence, unlike traditional Bayesian updating.

Contribution

It characterizes the conditions under which constriction of probability bounds occurs, expanding understanding beyond standard Bayesian updating.

Findings

Constriction can occur with evidence observed.

Constriction can occur without evidence.

Bayesian updating does not allow for constriction.

Abstract

Given a set of probability measures $\mathcal{P}$ representing an agent's knowledge on the elements of a sigma-algebra $\mathcal{F}$, we can compute upper and lower bounds for the probability of any event $A\in\mathcal{F}$ of interest. A procedure generating a new assessment of beliefs is said to constrict $A$ if the bounds on the probability of $A$ after the procedure are contained in those before the procedure. It is well documented that (generalized) Bayes' updating does not allow for constriction, for all $A\in\mathcal{F}$. In this work, we show that constriction can take place with and without evidence being observed, and we characterize these possibilities.