Pearl's and Jeffrey's Update as Modes of Learning in Probabilistic Programming
This work addresses a foundational problem in statistics and machine learning for researchers, but it is incremental as it builds on existing update mechanisms.
The paper tackles the relationship between Pearl's and Jeffrey's update rules in probabilistic programming, clarifying their similarities and differences through descriptions in terms of probabilistic programs, sampling semantics, and likelihood notions, and showing that Jeffrey's update arises via variational inference.
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.