The Magmoid of Normalized Stochastic Kernels

Normalization, $D(X + 1) \to D(X) + 1$, is almost a distributive law; but because one of the distributive law axioms only holds up-to-idempotent, it yields a non-associative composition of normalized kernels. We introduce the Markov magmoid of normalized stochastic kernels: a normalized-by-construction semantics for probabilistic inference, unifying exact Bayesian observations and interventions as two parenthesizations of the same composite. Front-door and back-door criteria follow from the axioms of Markov magmoids; we implement these with non-associative monadic notation.