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Method of Additional Structures on the Objects of a Monoidal Kleisli Category as a Background for Information Transformers Theory
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Category theory provides a compact method of encoding mathematical structures in a uniform way, thereby enabling the use of general theorems on, for example, equivalence and universal constructions. In this article we develop the method of additional structures on the objects of a monoidal Kleisli category. It is proposed to consider any uniform class of information transformers (ITs) as a family of morphisms of a category that satisfy certain set of axioms. This makes it possible to study in a uniform way different types of ITs, e.g., statistical, multivalued, and fuzzy ITs. Proposed axioms define a category of ITs as a monoidal category that contains a subcategory (of deterministic ITs) with finite products. Besides, it is shown that many categories of ITs can be constructed as Kleisli categories with additional structures.
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A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
Markov categories provide a synthetic, axiom-based framework in which conditional independence, sufficiency, completeness, and classical theorems such as Basu and Bahadur hold uniformly across many probability theories.
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