The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's update rule

TL;DR

This paper clarifies two main methods for updating beliefs with soft evidence in probabilistic reasoning, using a novel channel-based approach to unify and distinguish Jeffrey's and Pearl's update rules.

Contribution

It introduces a new channel-based framework to understand and compare Jeffrey's and Pearl's soft evidence update rules, highlighting their differences and applications.

Findings

Jeffrey's rule viewed as correction of beliefs

Pearl's method seen as an improvement process

Unified channel-based perspective enhances understanding

Abstract

Evidence in probabilistic reasoning may be 'hard' or 'soft', that is, it may be of yes/no form, or it may involve a strength of belief, in the unit interval [0, 1]. Reasoning with soft, [0, 1]-valued evidence is important in many situations but may lead to different, confusing interpretations. This paper intends to bring more mathematical and conceptual clarity to the field by shifting the existing focus from specification of soft evidence to accomodation of soft evidence. There are two main approaches, known as Jeffrey's rule and Pearl's method; they give different outcomes on soft evidence. This paper argues that they can be understood as correction and as improvement. It describes these two approaches as different ways of updating with soft evidence, highlighting their differences, similarities and applications. This account is based on a novel channel-based approach to Bayesian probability. Proper understanding of these two update mechanisms is highly relevant for inference, decision tools and probabilistic programming languages.