Towards a Categorical Account of Conditional Probability

Robert Furber, Bart Jacobs

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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.
Original languageEnglish
Title of host publicationProceedings of the 12th International Workshop on Quantum Physics and Logic (QPL 2015)
EditorsChris Heunen, Peter Selinger, Jamie Vicary
PublisherOpen Publishing Association
Number of pages17
Publication statusPublished - 4 Nov 2015
Event12th International Workshop on Quantum Physics and Logic - Oxford, United Kingdom
Duration: 13 Jul 201517 Jul 2015

Publication series

NameElectronic Proceedings in Theoretical Computer Science
PublisherOpen Publishing Association
ISSN (Electronic)2075-2180


Workshop12th International Workshop on Quantum Physics and Logic
Abbreviated titleQPL 2015
CountryUnited Kingdom
Internet address

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