This paper presents a categorical account of conditional probability,
covering both the classical and the quantum case. Classical conditional
probabilities are expressed as a certain "triangle-fill-in" condition,
connecting marginal and joint probabilities, in the Kleisli category of the
distribution monad. The conditional probabilities are induced by a map together
with a predicate (the condition). The latter is a predicate in the logic of
effect modules on this Kleisli category.
This same approach can be transferred to the category of C*-algebras (with
positive unital maps), whose predicate logic is also expressed in terms of
effect modules. Conditional probabilities can again be expressed via a
triangle-fill-in property. In the literature, there are several proposals for
what quantum conditional probability should be, and also there are extra
difficulties not present in the classical case. At this stage, we only describe
quantum systems with classical parametrization.Comment: In Proceedings QPL 2015, arXiv:1511.0118
