Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming

TL;DR

This paper explores the relationship between Pearl's and Jeffrey's probabilistic update rules, clarifying their differences and showing Jeffrey's rule arises from variational inference, using categorical probability theory.

Contribution

It provides a unified description of Pearl's and Jeffrey's updates through probabilistic programs and sampling semantics, and analyzes Jeffrey's rule via variational inference and category theory.

Findings

Jeffrey's update rule is derived via variational inference.

The paper clarifies the differences between Pearl's and Jeffrey's update mechanisms.

Categorical probability theory is used to analyze the update rules.

Abstract

The concept of updating a probability distribution in the light of new evidence lies at the heart of statistics and machine learning. Pearl's and Jeffrey's rule are two natural update mechanisms which lead to different outcomes, yet the similarities and differences remain mysterious. This paper clarifies their relationship in several ways: via separate descriptions of the two update mechanisms in terms of probabilistic programs and sampling semantics, and via different notions of likelihood (for Pearl and for Jeffrey). Moreover, it is shown that Jeffrey's update rule arises via variational inference. In terms of categorical probability theory, this amounts to an analysis of the situation in terms of the behaviour of the multiset functor, extended to the Kleisli category of the distribution monad.