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LIT-781: Equivalent Comparisons of Experiments

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Properties5
StatusActive — read 2026-10-09 (NOTE-595); worth reading as the paper in which the comparison of experiments is settled in general: for experiments with finitely many states, being at least as informative in every decision problem (Bohnenblust, Shapley and Sherman's order) and being able to reproduce the other by a stochastic transformation (Blackwell's 1951 sufficiency, later called garbling) are the same order. The finite-outcome converse is Sherman and Stein's, re-proved here by a minimax argument; the extension to arbitrary outcome spaces, by martingale convergence, is new. It adds comparison by k-action problems, and shows that for two states testing problems, and so the type I/type II error curves, decide the order.
Tagsmathematical-statistics, information-theory, mathematics
SupportsCLAIM-050 — For report fidelity, fidelity is relative to the communicative decisions receivers make, and Blackwell's order on rendering procedures, over a designed distribution of situations, makes precise the ceiling a rendering sets on those decisions, CLAIM-258 — Blackwell's order compares experiments about one parameter, so it does not reach the normal translation case: source and target readers decide about situations with changed coordinates, and where the state is common a translator with context knowledge yields a rendering that is not a garbling of the source, which the order typically leaves incomparable with it and about which the direction the manuscript uses is silent
Cited as “paper” byNOTE-595 — Equivalent Comparisons of Experiments
Cited as “source” byTHEORY-1096 — Blackwell's comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces, THEORY-1177 — For experiments on a finite parameter set, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation, THEORY-156 — For experiments on a finite parameter set whose outcome spaces are standard Borel, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation, THEORY-176 — A task-loss stitching penalty compares representations in one decision problem, through a fixed top network and a restricted map class, so it is not Blackwell's order: representations that are functions of each other can stitch differently, and a negative penalty shows the fitted map did not reproduce the layers it replaced, THEORY-717 — For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix

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David Blackwell (1953), The Annals of Mathematical Statistics 24(2):265–272 — DOI-10.1214/aoms/1177729032

Key takeaways

  • The equivalence, and whose it is. “More informative” (α ⊃ β: every risk vector attainable with β is attainable with α, for every closed bounded convex set of loss vectors) and “sufficient for” (α ≻ β: a stochastic transformation carries α’s distributions to β’s under each of the n states) are the same order. Blackwell (1951) had ≻ ⇒ ⊃ (Theorem 3 here), and Sherman and Stein the converse for finitely many outcomes. This paper re-proves the finite case as a saddle-point statement about h(D, M) = Trace((Q − PM)D) (Theorem 6) and proves the general case (Theorem 8). The word “garbling” does not occur in it.
  • How the general case goes. Both orders reduce to the experiments’ standard measures, the laws of the normalised likelihood vector on the simplex (Theorems 1 and 2, quoted from Bohnenblust, Shapley and Sherman and from Blackwell 1951). There ⊃ is the convex order, ≻ is a mean-preserving transformation, and Theorem 8 shows the first implies the second on any bounded subset of ℝⁿ: dyadic approximations on finite sets, the finite theorem, and Doob’s martingale convergence.
  • k-decision problems (Theorem 9). Restricting to problems with k actions gives a weaker order ≻k with three equivalent forms: every k-outcome experiment reproducible from β is reproducible from α; inclusion of risk sets over k-point problems; ∫ φ dm_α ≥ ∫ φ dm_β for maxima of k linear functions. ≻{k+1} implies ≻_k, and ≻_k for every k gives ≻; that each is strictly stronger than the last is cited to unpublished work of Stein.
  • Dichotomies (Theorems 10–11). With two states, ≻_2 already gives ≻, so α ≻ β iff α’s least type II error at every level is at most β’s; and then, since ≻ passes to repeated sampling, at every sample size.

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 on 2026-10-09 (NOTE-595). The reading is the source of THEORY-156, the record’s statement of the equivalence and its scope. The book-length treatment, with Le Cam’s deficiency as the quantitative version, is LIT-778.

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