Proposition [lcc-000O]

Consider The category of random morphisms for some Markov category \mathcal {C}

  • There is an obvious functor \mathsf {RandMor}(\mathcal {C}) \to \mathcal {C}, which carries (\mu : I \to \Omega , f_1: \Omega \otimes X \to Y) to the composite f_1 (\mu \otimes X).

  • That functor is full if and only if \mathcal {C} has randomness pushback in the sense of A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics, definition 11.19
  • If \mathcal {C} has randomness pushback, then given \phi : \Omega _1 \to \Omega _2 witnessing the equivalence (\Omega _1, \mu _1, f_1) \sim (\Omega _2, \mu _2, f_2), there exist a further tuple (\Omega , \mu , f) with deterministic \psi _1: \Omega \to \Omega _1, \psi _2: \Omega \to \Omega _2 witnessing its equivalence to both morphisms. In particular, in this case it suffices to quotient the set of tuples by the equivalence relation generated by deterministic morphisms.

Proof

The functoriality is trivial to verify. Fullness of this functor is just a reformulation of the definition of randomness pushback.

Given \phi : \Omega _1 \to \Omega _2, let \phi ': \Omega _1 \otimes A \to \Omega and \nu : I \to A be a randomness pushback for it---that is, \phi ' is deterministic and \phi ' (\Omega _1 \otimes \nu ) = \phi . Consider the tuple (\Omega _1 \otimes A, (\mu _1 \otimes \nu ), f_1(\pi _{\Omega _1} \otimes X)). By construction, the deterministic maps \pi _1 and \phi ' witness its equivalence to the two other maps.

References

Reference A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics [fritz-synthetic-markov-cats]

Abstract

  • August , 2020
  • Tobias Fritz

We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; various versions of conditional independence and its standard properties; conditional products; almost surely; sufficient statistics; as well as versions of theorems on sufficient statistics due to Fisher-Neyman, Basu, and Bahadur. Besides the conceptual clarity offered by our categorical setup, its main advantage is that it provides a uniform treatment of various types of probability theory, including discrete probability theory, measure-theoretic probability with general measurable spaces, Gaussian probability, Markov processes of either of these kinds, and many others. 

 @article
{fritz-synthetic-markov-cats, title={A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics}, volume={370}, ISSN={00018708}, DOI={10.1016/j.aim.2020.107239}, abstractNote={We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; various versions of conditional independence and its standard properties; conditional products; almost surely; sufficient statistics; as well as versions of theorems on sufficient statistics due to Fisher-Neyman, Basu, and Bahadur. Besides the conceptual clarity offered by our categorical setup, its main advantage is that it provides a uniform treatment of various types of probability theory, including discrete probability theory, measure-theoretic probability with general measurable spaces, Gaussian probability, Markov processes of either of these kinds, and many others.}, url={https://arxiv.org/abs/1908.07021}, journal={Advances in Mathematics}, author={Fritz, Tobias}, year={2020}, month={Aug}, pages={107239},language={en} }

Related

Definition The category of random morphisms [lcc-000N]

Let \mathcal {C} = (\mathcal {C}, \otimes , I) be a Markov category. Then the category of random morphisms in \mathcal {C}, denoted \mathsf {RandMor}(\mathcal {C}) is defined as follows:

  • The objects are the objects of \mathcal {C}.
  • A morphism X \to Y is given by an object \Omega \in \mathcal {C}, a state I \to \Omega and a deterministic map X \otimes \Omega \to Y, up to the equivalence relation that identifies (\Omega _1, \mu _1, f_1) \sim (\Omega _2, \mu _2, f_2) if there exists a map \phi : \Omega _1 \to \Omega _2 so that \phi \mu _1 = \mu , f_1(X \otimes \phi ) = f_2
  • The identity on X is (I, 1_I, \lambda _x: I \otimes X \cong X)
  • Composition is given by (\Omega _1, \mu _1, f_1) \circ (\Omega _2, \mu _2, f_2) = (\Omega _1 \otimes \Omega _2, \mu _1 \otimes \mu _2, f_1 \circ (\Omega _1 \otimes f_2)) (the only thing that makes sense)