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LIT-772: A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics

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StatusActive — read 2026-10-09 (NOTE-588); worth reading as the reference account of Markov categories and of how much probability and statistics follows from copying, discarding and composing kernels: determinism, four notions of conditional independence with the semigraphoid rules (weak union only with conditionals), almost-sure equality, and sufficiency, completeness and ancillarity. Basu's theorem holds in every Markov category, and Fisher–Neyman and Bahadur under strict positivity, which Stoch has without conditionals. The abstract Fisher–Neyman is matched to the classical theorem only for finite sets.
Tagsmathematics, mathematical-statistics, probabilistic-modeling
Cited as “paper” byNOTE-588 — Fritz — A synthetic approach to Markov kernels
Cited as “source” byTHEORY-157 — The classical theorems on sufficient statistics need only the copy/discard structure of Markov kernels: Basu's theorem holds in every Markov category, and Fisher–Neyman and Bahadur need only strict positivity, not the existence of conditional distributions

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Tobias Fritz (2020), Advances in Mathematics 370:107239 — ARXIV-1908.07021

Key takeaways

  • The definition (Definition 2.1). A Markov category is a symmetric monoidal category with a commutative copy/discard comonoid on every object, compatible with ⊗, and with discard natural, so the unit is terminal. Copy is deliberately not natural. The morphisms for which it is are the deterministic ones (Definition 10.1). Examples: FinStoch, Stoch, BorelStoch, Gauss (affine maps plus Gaussian noise), the Radon Kleisli category, and stochastic processes as diagram categories (Section 7).
  • Optional axioms (Section 11): conditionals, randomness pushback, positivity and causality. FinStoch, BorelStoch and Gauss have conditionals. Stoch does not, but is positive, strictly positive and causal. Conditionals are unique only in a meet-semilattice (Proposition 11.15), and unique a.s. in general (Proposition 13.7).
  • Conditional independence without conditionals (Section 12). The paper has four notions: conditioning on an output, on an input, on both, and the Markov-chain form A ⊥⊥ Y | X. Symmetry, decomposition and contraction hold everywhere; weak union is proved only with conditionals. With conditionals, Dawid–Studený conditional products exist and are unique (Proposition 12.9).
  • Almost surely (Section 13). Probability spaces and kernels modulo a.s. equality form a symmetric monoidal category (Proposition 13.9). With conditionals, Bayesian inversion is a dagger functor on it (Remark 13.10).
  • Statistics (Sections 14–16). A statistic is a deterministic morphism. Sufficiency means the sample can be regenerated from the statistic independently of θ. Completeness is a property of the kernel sp; in Stoch it is bounded completeness. Basu (Theorem 15.8) holds in every Markov category. Fisher–Neyman (14.5) and Bahadur (16.3) need strict positivity. Fisher–Neyman is shown to be the classical factorisation in FinStoch only.
  • Signed probabilities (Example 11.27). FinStoch± satisfies the Markov axioms but is not positive and has no conditionals.

Standing in the record

Filed on 2026-10-09 at the owner’s request, as one of the works in the reference list of the owner’s working manuscript (October 2026) that the record did not yet hold. See the curation entry of that day.

Read the same day (NOTE-588). Its result on which axioms the sufficiency theorems use is filed as THEORY-157. Within the record, Fritz’s earlier convex spaces enter the reading of LIT-038 (NOTE-050), and Baez–Fritz–Leinster’s entropy characterisation is cited in NOTE-088. Its example of signed kernels (FinStoch±) shows where negativity enters, against THEORY-012’s finding that signed global sections always exist. The paper does not draw that connection; it is mine. Baez and Fong’s Noether theorem for Markov processes (LIT-773), filed with it, works with the same kernels as operators and declares no relation.

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