  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.
