Random probability distributions as natural transformations
summary
Claim: homogeneous random distributions — where atom locations are independent of atom weights — bijectively correspond to natural transformations G -> GG for G the Giry monad. The work formalizes polymorphic programs of type Prob A -> Prob A as these natural transformations, uses category theory to prove every such transformation decomposes into a discrete weighted part plus a base measure part, and connects to examples like Dirichlet processes and Kingman’s paintbox construction while noting applicability to probabilistic programming.
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