Learning from What's Right and Learning from What's Wrong

TL;DR

This paper clarifies the mathematical distinction between Pearl's and Jeffrey's rules for updating probability distributions, framing them as learning from what's right versus what's wrong, with implications for machine learning and predictive coding.

Contribution

It provides the first precise mathematical characterization of the difference between Pearl's and Jeffrey's updating rules, linking them to validity and divergence minimization.

Findings

Pearl's rule increases validity (expected value).

Jeffrey's rule decreases Kullback-Leibler divergence.

The distinction is illustrated through a cognitive scenario.

Abstract

The concept of updating (or conditioning or revising) a probability distribution is fundamental in (machine) learning and in predictive coding theory. The two main approaches for doing so are called Pearl's rule and Jeffrey's rule. Here we make, for the first time, mathematically precise what distinguishes them: Pearl's rule increases validity (expected value) and Jeffrey's rule decreases (Kullback-Leibler) divergence. This forms an instance of a more general distinction between learning from what's right and learning from what's wrong. The difference between these two approaches is illustrated in a mock cognitive scenario.