Note of 2026-10-27: the easy direction split off
The status review of this date (ADR-057) split the definitions and the easy direction into THEORY-1096, Active, with its proof. Ten of this account’s twelve dependents used nothing more and moved there. CLAIM-259 and CLAIM-326 use the converse and stay grounded here. The blocker is unchanged: a second proof of the general converse.
Source
Blackwell (1953), LIT-781, Theorems 1–8, as read in NOTE-595.
What was actually shown
An experiment is an n-tuple of probability measures on a common space, one per state. α is “more informative” than β (α ⊃ β) if, for every closed bounded convex set A of loss vectors, every risk vector attainable with β is attainable with α. α is “sufficient for” β (α ≻ β) if there is a stochastic transformation T with T m_i = M_i for every state i: β is a garbling of α.
≻ implies ⊃ by Jensen’s inequality (Theorem 3, from Blackwell 1951). For finite outcome spaces the converse is a minimax argument: the defect Q − PM of the best Markov matrix M costs nothing against any decision matrix, so it vanishes (Theorem 6, Sherman–Stein, new proof). For arbitrary outcome spaces, both orders reduce to a comparison of standard measures on the simplex (Theorems 1 and 2, quoted), where ⊃ is the convex order and ≻ is a mean-preserving transformation; Theorem 8 proves convex order implies such a transformation on any bounded subset of ℝⁿ by finite approximation and martingale convergence. The support is proof, not experiment: what could show it wrong is an error in a step, so the promotion condition asks for a checked second proof, and for the two quoted results to be checked at their source.
For two states the order is decided by testing problems alone: α ≻ β iff α’s least type II error at every level t is at most β’s (Theorem 10 and its Corollary).
What this does not say
- Not a measure. Most pairs of experiments are incomparable; the theorem says nothing about how much information is lost when neither simulates the other. That quantitative question is Le Cam’s deficiency (LIT-778, not read).
- Not for infinitely many states, and not for unbounded losses, as proved here.
- Not, as re-derived here, for outcome spaces that are not standard Borel. The 1953 paper claims them through Theorem 2, quoted from Blackwell (1951), which the record has not read (QUESTION-321).
- Not “finitely many decision problems suffice”. The k-action comparisons ≻_k are weaker than ≻ in general (asserted from Stein’s unpublished work); only for two states does k = 2 suffice.
- Not that the simulating kernel is constructive or unique; Theorem 8’s kernel comes from a compactness and limit argument.
- Not a statement about any particular informativeness functional. That a quantity such as mutual information or an f-divergence is monotone under garbling is a consequence one may draw, but the paper states only the convex-function criterion of Theorem 1.
Note of 2026-10-19: the finite case filed apart
The finite-outcome case is THEORY-717, Active. The paper proves it without the quoted results: Theorems 4–6 for the converse, and a one-line matrix argument for the direct implication. Claims that cite this THEORY for experiments with finitely many outcomes can cite that one instead. This one is Proposed only for the arbitrary-outcome extension.
The citing claims were checked the same day. CLAIM-323, whose case is finite, now rests on that THEORY alone. CLAIM-237 and CLAIM-326 cite both, each with a note saying when the finite case suffices. The other ten use the definition or the order for unrestricted renderings, and they stay here.
Note of 2026-10-29: the general converse, re-derived
The twelfth status review re-derived the converse for any α and a β whose outcome space is standard Borel, without the 1951 results the 1953 paper quotes. Write μ and ν for the standard measures of α and β, the laws of the posterior p under the uniform prior, on the simplex Δ.
- Bayes risks give the convex order on max-affine functions. For a finite problem with losses L(a, i) and the uniform prior, the Bayes risk is ∫ min_a ⟨L_a, p⟩ dμ(p). If α does at least as well as β in every such problem, then ∫ φ dμ ≥ ∫ φ dν for every φ = max_j (⟨c_j, p⟩ + d_j).
- A martingale coupling exists (the content of Theorem 8). Discretize ν by conditional expectation on a finite partition of Δ into cells of diameter at most 1/k: ν_k = Σ_j w_j δ_{y_j}, with ν_k ≤cx ν by Jensen’s inequality and ν_k → ν. Then μ splits as Σ_j w_j μ_j with the barycentre of μ_j equal to y_j. The splittings, densities f_j ≥ 0 summing to 1 in L∞(μ), form a weak* compact convex set; their image under f ↦ (∫ f_j dμ, ∫ p f_j dμ)_j is compact and convex in finite dimensions; and a functional (c_j, d_j) separating the target from it would give Σ_j w_j (⟨c_j, y_j⟩ + d_j) > ∫ max_j (⟨c_j, p⟩ + d_j) dμ ≥ ∫ max_j (⟨c_j, p⟩ + d_j) dν_k ≥ Σ_j w_j (⟨c_j, y_j⟩ + d_j), a contradiction. So there are couplings π_k of ν_k and μ with ∫ g(y)(x − y) dπ_k = 0 for every continuous g. On the compact Δ × Δ a subsequence converges, the condition passes to the limit, and the limit π couples ν and μ with E[X | Y] = Y.
- The coupling gives a garbling. Under state i, α’s posterior has law n x_i μ(dx), and β’s has law n y_i ν(dy). Let K be π’s conditional law of Y given X, and C(q, ·) the conditional law of β’s outcome given its posterior q, which exists as a kernel because β’s outcome space is standard Borel, and is the same under every state because the posterior is sufficient. Then T(x, E) = ∫ K(p_α(x), dq) C(q, E) satisfies ∫ T(x, E) m_i(dx) = n ∫ q_i C(q, E) ν(dq) = M_i(E), using E[X_i | Y = q] = q_i.
Step 3 is where standard Borel is used; arbitrary spaces are QUESTION-321. The check (numpy, run by the reviewing session, on three-state Gaussian experiments on a 400-point grid, with β a Gaussian garbling of α): step 3’s identities hold to 3 × 10⁻¹⁵, with T built from the coupling alone; step 2’s splitting residual falls from 8.3 × 10⁻³ at 10³ Frank–Wolfe iterations to 1.1 × 10⁻³ at 6.4 × 10⁴, while a pair not in convex order stays at 5.5 × 10⁻²; and the discretization error falls from 0.208 at 4 cells to 0.008 at 120. The proof is the evidence; the numbers check its finite steps.