The claim
Setting. U is the source and V the rendering, both experiments about one situation Z with finitely many values. U and V each take finitely many values. That is the case once renderings are coded into finitely many categories. A decision problem q has a finite set of actions and a loss L_q(z, a) in [0, 1], and π is a stated prior on Z. r_q,π(U) is the least expected loss under π over rules that read U, and
D⁺_Q(U → V; π) = sup over q ∈ Q of [r_q,π(V) − r_q,π(U)]⁺.
(a) A garbling never gains. Suppose P(V | Z) = G P(U | Z) for a Markov matrix G. A rule δ that reads V gives the rule “apply G to U, then δ”, with the same distribution of actions under each z. So r_q,π(U) ≤ r_q,π(V), and D⁺(V → U; π) = 0.
(b) The converse. Suppose D⁺(U → V; π) = 0 for every finite problem with losses in [0, 1] and every prior. Then r_q,π(V) ≤ r_q,π(U) for all of them. Rescaling a bounded loss by L ↦ (L − m)/(M − m) maps every Bayes risk by the same increasing affine map, so the inequality holds for every bounded loss. By Blackwell’s Theorem 6 (THEORY-717), U is then reproducible from V by a Markov matrix.
(c) It is directed. With Z uniform on {0, 1}, U = Z, V constant and q = “guess Z” under 0–1 loss, r(U) = 0 and r(V) = 1/2. So D⁺(U → V) = 1/2 and D⁺(V → U) = 0.
(d) A86’s D_Q is its symmetrization. With the identity on tasks and the same prior, D_Q(U, V) = sup_q |r_q(U) − r_q(V)|. Since |x| = max([x]⁺, [−x]⁺), and a supremum of maxima is the maximum of the suprema, D_Q(U, V) = max(D⁺(U → V), D⁺(V → U)).
For example, take CLAIM-323’s experiments: Z is uniform on {1, 2, 3}, U says whether Z = 1, and V says whether Z = 3.
- On “is Z = 3?” the risks are 1/3 from U and 0 from V. On “is Z = 1?” they are 0 and 1/3.
- So the directed pair is (1/3, 1/3).
- A constant rendering has risk 1/3 on both problems, which gives the pair (1/3, 0).
- D_Q is 1/3 in both cases.
What it does not say
- It says nothing about outcome spaces that are not finite, such as raw texts. There step (b) needs the general converse, which CLAIM-326 keeps on THEORY-156.
- It does not say this is the decision term fidelity should use. That is CLAIM-259’s proposal.
- It does not state the prior-free form or the relation to Le Cam’s deficiency.