Updating Probabilities

TL;DR

This paper investigates when naive probability updates are valid, focusing on the CAR condition and its limitations, and explores generalized updating methods like Jeffrey conditioning and MRE, revealing their applicability and constraints.

Contribution

It generalizes the CAR condition to Jeffrey conditioning and shows the absence of such conditions for MRE, connecting previous results in probability updating.

Findings

CAR condition holds infrequently in practice

Jeffrey conditioning has specific criteria for correctness

No general conditions exist for MRE to be appropriate

Abstract

As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a ``naive space', which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR (coarsening at random) in the statistical literature characterizes when ``naive' conditioning in a naive space works. We show that the CAR condition holds rather infrequently. We then consider more generalized notions of update such as Jeffrey conditioning and minimizing relative entropy (MRE). We give a generalization of the CAR condition that characterizes when Jeffrey conditioning leads to appropriate answers, but show that there are no such conditions for MRE. This generalizes and interconnects previous results obtained in the literature on CAR and MRE.