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

Bart Jacobs, Dario Stein

TL;DR

This work investigates the relationship between Pearl's update and Jeffrey's update within probabilistic reasoning and probabilistic programming. It formalizes both updates in WebPPL, introduces two distinct likelihood notions, and shows that Jeffrey's rule arises from variational inference when the multiset functor $\mathcal{M}$ is extended to the Kleisli category of the distribution monad $\mathcal{D}$. Key results connect frequentist learning and variational approaches to Jeffrey's update, provide operational rejection-sampler interpretations, and prove a new commutation property between the extended multiset functor and channel daggers. The paper thus clarifies a population-versus-individual learning perspective and delivers a rigorous, implementable framework for updates in probabilistic programming.

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.

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

TL;DR

This work investigates the relationship between Pearl's update and Jeffrey's update within probabilistic reasoning and probabilistic programming. It formalizes both updates in WebPPL, introduces two distinct likelihood notions, and shows that Jeffrey's rule arises from variational inference when the multiset functor is extended to the Kleisli category of the distribution monad . Key results connect frequentist learning and variational approaches to Jeffrey's update, provide operational rejection-sampler interpretations, and prove a new commutation property between the extended multiset functor and channel daggers. The paper thus clarifies a population-versus-individual learning perspective and delivers a rigorous, implementable framework for updates in probabilistic programming.

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.
Paper Structure (12 sections, 11 theorems, 28 equations, 2 figures)

This paper contains 12 sections, 11 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Motivating Example
  3. Multisets, Distributions, and Channels
  4. Validity, Conditioning, and Pearl's Update Rule
  5. Dagger channels and Jeffrey's update rule
  6. An Operational Understanding of Jeffrey's Update
  7. Likelihoods and Generative Models for Pearl and Jeffrey
  8. Jeffrey's Update Rule via Variational Inference
  9. Jeffrey's rule via Frequentist Learning
  10. Jeffrey's Rule as Variational Inference
  11. Conclusions
  12. Appendix: Overview of WebPPL

Key Result

lemma 1

For a channel $c\colon X \mathrel{\ooalign{$→$\cr \!$\begin{tikzpicture}[baseline=-2.6pt] \filldraw[fill=white,draw=white] circle (1.0pt); \filldraw[fill=white,draw=black,line width=0.2pt] circle (1.2pt); \end{tikzpicture}$\cr}} Y$, a distribution $\omega\in\m

Figures (2)

  • Figure 1: Ticker device
  • Figure 2: Graphical representation of Jeffrey likelihood on the left, and Pearl likelihood on the right, see Definition \ref{['LikelihoodDef']}.

Theorems & Definitions (12)

  • lemma 1
  • theorem 1
  • theorem 2
  • lemma 2
  • definition 1
  • theorem 3
  • theorem 4
  • lemma 3
  • proposition 1
  • theorem 5
  • ...and 2 more