From probability monads to commutative effectuses

Abstract

Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These 'probabilistic' models are called commutative effectuses, and are the focus of attention here. The paper describes the main known 'probability' monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative effectus. The main properties are: partial additivity, strong affineness, and commutativity. It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate-and state-transformers, normalisation and conditioning of states.

On algebraic abstractions for concurrent separation logics

“…Interestingly, the notion of homomorphism for effect algebras [Cho 2015;Jacobs 2018] is similar to our notions of PCM morphisms. Indeed, the similarities between our Definition 3.4 and Definition 3.6 with [Cho et al 2015, Definition 12] are clear.…”

“…We aren't aware of any other work that considers separating relations as a standalone concept. That said, the key separating relation property of associativity (property 5 in Definition 3.4) has been considered before [Jacobs 2018;Krebbers 2015], though as a property of the disjointness relation ⊥ of the underlying PCM. In our setting, the latter is just one possible separating relation, associated with total PCM morphisms.…”

Denotational validation of higher-order Bayesian inference

“…We now briefly recount the relevant parts of Kock Kock's [20122012] development. (In the finite discrete case, this is also related to Jacobs Jacobs's [2017] work on effectuses.) 4.2.1 Axioms and structure.…”

A channel-based perspective on conjugate priors

“…For those with categorical background knowledge: we will be working in the Kleisli categories of the distribution monad D for discrete probability, and of the Giry monad G for continuous probability, see e.g. Giry (1982), Panangaden (2009) and Jacobs (2018). Discrete distributions may be seen as a special case of continuous distributions, via a suitable inclusion map D → G. Hence one could give one account, using G only.…”