Source
Fritz (2020), LIT-772, read in NOTE-588: Definitions 13.1, 13.16, 14.1–14.3, 15.1, 15.7, 16.1–16.2; Theorems 14.5, 15.8, 16.3; Examples 13.19, 14.6, 15.4.
What was actually shown
A Markov category is a symmetric monoidal category with a commutative copy/discard comonoid on each object and natural discard. In one, Fritz defines:
- a statistic: a deterministic morphism s : X → V for a model p : Θ → X;
- sufficiency: there is α : V → X such that, given s, the sample can be regenerated by α independently of θ;
- completeness: of the kernel sp, gf = hf implies g = h f-a.s.;
- ancillarity: sp does not depend on θ.
He then proves three theorems:
- Basu (Theorem 15.8): a complete sufficient statistic and an ancillary statistic are independent given θ, with no extra axiom.
- Fisher–Neyman (Theorem 14.5): under strict positivity, s is sufficient iff some α has αsp = p and sα = id sp-a.s.
- Bahadur (Theorem 16.3): under strict positivity, if a minimal sufficient statistic exists, any complete sufficient statistic is minimal.
Strict positivity follows from the existence of conditionals (Lemma 13.18). It also holds in Stoch directly (Example 13.19), and Stoch lacks conditionals (Example 11.18). So none of the three theorems needs regular conditional distributions.
The definitions could have failed to match the classical ones, and Fritz checks them. Abstract completeness in Stoch is bounded completeness (Example 15.4). In FinStoch the abstract Fisher–Neyman condition is the factorisation p(x|θ) = g_θ(s(x)) h(x) (Example 14.6).
What this does not say
- Not that Markov categories prove anything new about classical statistics. The content is about hypotheses: which structure each proof uses. Classical Basu already holds with bounded completeness.
- Not that the measure-theoretic Fisher–Neyman theorem follows. Fritz leaves that open (end of Section 14). In Stoch, Theorem 14.5 is a theorem about his notion of sufficiency, and that notion is not shown to agree with the classical one there.
- Not that the abstract notions are final. Fritz calls all his definitions provisional (Section 1). His a.s. equality in Stoch is weaker than pointwise a.s. equality on σ-algebras that are not countably generated (Example 13.3, after Fremlin). So “sufficient” in Stoch means sufficient in that weaker sense.
- Not that positivity is dispensable. The Fisher–Neyman and Bahadur proofs use it. FinStoch± (Example 11.27) satisfies the Markov axioms without positivity, and the paper does not say whether the theorems fail there.
- Not that statistics in general needs no measure theory. The proofs that particular categories (Stoch, BorelStoch) satisfy the axioms carry the measure-theoretic work. Fritz says so (Section 1).