The Giry Monad: Randomized Morphisms

Abstract

Morphisms relate stochastic relations in a way that preserves the probabilistic structure. We did define a morphism f : K → L between stochastic relations K = ((X,A), (Y, B),K) and L = ((A,D), (B, ε), L) as a pair (f, g) of surjective measurable maps so that 𝔖(g) ◦ K = L ◦ f. This means that for each measurable subset F of B, and for each x ∈ X the equality K(x)(g ⁻¹ [F])= L(f(x))(F) holds. Stochastic relations form a category with these morphisms, and the kernels of morphisms are exactly the congruences; see Section 1.7.3.