Coarse-graining and compounding as monads

Two very basic constructions involving experimental procedures are the formation of coarse-grained versions of experiments, and the formation of branching sequential experiments. The latter allow for the conditioning of states on the results of previous measurements. When one conditions on the results of different coarse-grainings of the same previous experiment, the possibility of interference effects arises. Here, I show how to formulate both constructions in terms of monads on a suitable category of (general) probabilistic models. Moreover, I show that these are connected by distributive law, allowing for a composite monad describing the closure of a probabilistic model under both coarse-graining and sequential measurement. Algebras for all three monads are characterized; lessons are drawn regarding the possibility of interference and also regarding the formation of sequential products of effects; and connections are made with some themes from the older quantum-logical literature.