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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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Properties5
StatusActive
Tagsmathematical-statistics, information-theory, mathematics
SourceLIT-781 — Equivalent Comparisons of Experiments
Extended byTHEORY-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
SupportsCLAIM-237 — The first open problem of directed transport is not yet posed: a contextual model is a menu of incompatible experiments, and under the natural order on menus any state-independent classical simulation is dominated automatically, so what is open is only the converse or a deficiency bound, CLAIM-323 — Where source and rendering are experiments about one common situation, a rendering that depends on the situation other than through the source is not a garbling of it and can be incomparable with it in Blackwell's order: with Z uniform on {1, 2, 3}, a source that tells whether Z = 1 and a rendering that tells whether Z = 3 each have the lower 0–1 risk on one of the problems of saying whether Z = 1 and whether Z = 3, so neither dominates, CLAIM-326 — With losses in [0, 1] and a stated prior, the positive part of the Bayes-risk increase on the same decision problem, D⁺_Q(U → V; π), is directed, is zero from a garbling back to its source, vanishes for every finite problem and prior exactly when the rendering is at least as informative as the source in Blackwell's order, over finitely many states and a source whose outcome space is standard Borel, and has A86's D_Q, with the identity on tasks, as its symmetrization, CLAIM-380 — With finitely many states and finitely many outcomes, the positive part of the Bayes-risk increase on the same decision problem is directed, is zero from a garbling back to its source, vanishes for every finite problem and prior exactly when the rendering is at least as informative as the source, and has A86's D_Q as its symmetrization, CLAIM-381 — Over a finite class of items with finite judgement outcomes, a constant outcome kernel fits every item only when the target readers' averaged responses do not vary with the item, an exact fit is a garbling with the item as common parameter, and Le Cam's deficiency and its restriction to finitely many decision problems are finite computations, CLAIM-409 — For one state law, one intervention distribution and one outcome, a deficiency of sufficiency cannot increase under refinement: if T₁ = g∘T₂ then Γ(T₂) ≤ Γ(T₁); informational closure, by contrast, is monotone in neither direction, since in the three-bit chain of Rosas et al. (2024) A is closed, its refinement (A, B) is not and the full state refining that is closed again, while A⊕B, a coarsening of (A, B), is not closed either; closure fails to be monotone because refining a description changes the future it must predict as well as the evidence, so Blackwell dominance, which every deterministic refinement has over its coarsening, does not order closure

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Source

Blackwell (1953), LIT-781, Theorems 4–6, as read in NOTE-595 (its claim C1, “strong (proof)”). The finite converse is Sherman and Stein’s (1951), and the paper gives a new proof of it.

What was actually shown

An experiment with n states and N outcomes is an n × N Markov matrix P, whose row i is the outcome distribution under state i. A decision rule with k actions is an N × k Markov matrix D, and with losses L its risk in state i is (PDLᵀ)ᵢᵢ. P ⊃ Q (P is at least as informative) when every risk vector attainable with Q is attainable with P, for every decision problem. P ≻ Q when Q = PM for a Markov matrix M (Theorem 5).

  • ≻ implies ⊃. If Q = PM and D is a rule for Q, then MD is a rule for P, and PMD = QD, so it attains the same risks. The record writes this out directly, without the paper’s Theorem 3, which runs through the quoted Theorems 1–2.
  • ⊃ implies ≻ (Theorem 6). The bilinear function h(D, M) = Trace((Q − PM)D), over Markov matrices M and matrices D with entries in [0, 1], has a saddle point (Bohnenblust, Karlin and Shapley). Theorem 4 uses P ⊃ Q to make its value at most 0. So, at the saddle point, U = Q − PM₀ has no positive entry. Its entries sum to 0, because Q and PM₀ are both Markov, so U = 0.

The support is a proof. What could show it wrong is an error in a step, and NOTE-595 followed each step.

What this does not say

  • Anything for infinite outcome spaces. That case is Theorem 8, which rests on Theorems 1–2, quoted from 1951 and not proved in the paper. It is THEORY-156, which is Proposed until those results are checked or a second proof is read.
  • Not a measure. It decides whether one experiment can simulate the other. When neither can, it says nothing about how much is lost, which is Le Cam’s deficiency (LIT-778, not read).
  • Not for infinitely many states.
  • Not that M is unique or explicit. M comes out of the saddle point.
  • Not a statement about a particular functional. That mutual information or an f-divergence can only fall under garbling follows from ≻. It is not the order.

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