The claim
Setting. Θ, X and Y are finite. The experiment M is (P_θ) with P_θ ∈ Δ(X), and N is (Q_θ) with Q_θ ∈ Δ(Y). A kernel K sends X to Δ(Y), and (KP_θ)(y) = Σ_x K(y | x) P_θ(x). TV(p, q) = ½ Σ |p − q| = max over 0 ≤ h ≤ 1 of ⟨p − q, h⟩.
- δ := min_K max_θ TV(KP_θ, Q_θ). This is CLAIM-381’s linear programme, and the minimum is attained, the kernels forming a compact set.
- D := the supremum, over finite action sets A, losses L : Θ × A → [0, 1] and rules δ_N : Y → Δ(A) for N, of min over rules δ_M for M of max_θ [R_θ(δ_M) − R_θ(δ_N)], where R_θ(δ_M) = Σ_x P_θ(x) Σ_a δ_M(a | x) L(θ, a).
M is ε-deficient relative to N, in the decision sense of Han’s slide 11 (LIT-1847), exactly when D ≤ ε. The claim is D = δ.
(i) D ≤ δ. Take K* attaining δ, and for any A, L and δ_N put δ_M = δ_N ∘ K*, that is δ_M(a | x) = Σ_y δ_N(a | y) K*(y | x). Then R_θ(δ_M) − R_θ(δ_N) = ⟨K*P_θ − Q_θ, h_θ⟩ with h_θ(y) = Σ_a δ_N(a | y) L(θ, a) ∈ [0, 1], so the gap is at most TV(K*P_θ, Q_θ) ≤ δ, at every θ. This is Han’s slide 14.
(ii) δ ≤ D. Fix the problem A = Y with δ_N the identity. A rule for M is then a kernel K, and its gap at θ is ⟨KP_θ − Q_θ, L(θ, ·)⟩.
- Let G = {g on Θ × Y : 0 ≤ g(θ, y) ≤ π(θ) for some π ∈ Δ(Θ)}, a compact convex polytope, and f(K, g) = Σ_θ Σ_y g(θ, y)[(KP_θ)(y) − Q_θ(y)], bilinear in K and g.
- (a) Each g ∈ G is πL with L ∈ [0, 1]: put L = g/π where π(θ) > 0 and 0 elsewhere. For every K, f(K, g) = Σ_θ π(θ) gap_θ(K) ≤ max_θ gap_θ(K). So min_K f(K, g) ≤ min_K max_θ gap_θ(K) ≤ D, and max_g min_K f ≤ D.
- (b) The stochastic matrices and G are non-empty compact convex sets in finite dimensions, and f is bilinear, so by von Neumann’s minimax theorem min_K max_g f = max_g min_K f.
- (c) For fixed K and π, the best L(θ, y) is 1[(KP_θ − Q_θ)(y) > 0], so max_{g ∈ G} f(K, g) = max_π Σ_θ π(θ) TV(KP_θ, Q_θ) = max_θ TV(KP_θ, Q_θ).
Hence δ = min_K max_g f = max_g min_K f ≤ D, and with (i), D = δ. The supremum defining D is attained, at A = Y, δ_N the identity and the loss L* = g*/π* from a maximising g*. For losses in [0, c] the bound scales to cε. At ε = 0 this is the finite Blackwell–Sherman–Stein theorem (THEORY-717), a check on the statement.
The step Han’s slide 15 leaves as “swap the inf and sup” is (b), with the conditions it needs: finite outcome sets, so that Y is an admissible action set and the rules form a compact convex set, and the bilinear form in the product π(θ)L(θ, a). van Rooyen’s Theorem 26 (LIT-1859, NOTE-1430) proves the converse for one fixed prior; the uniform form follows from it by exchanging the garbling with the prior, which is (b) and (c) again.
Checked numerically. p21/claim488_lecam_converse.py (numpy only, with its own simplex) solves CLAIM-381’s programme and, separately, the programme for max_g min_K f. On 60 random experiments with |Θ|, |X| and |Y| from 2 to 4 they agree to 1.1 × 10⁻¹³, and on the single problem built from g* no rule for M does better than δ uniformly. Its controls: known answers (an uninformative experiment against the identity gives 0.5; an exact garbling gives 0), the returned kernel recomputed directly, 20,000 random kernels that never beat the programme, 600 random problems in which no optimal gap exceeds δ, and two negative controls (dropping the ½ is detected, and van Rooyen’s bound fails for a signed loss as NOTE-1430 says). The uniform-prior average falls strictly below δ in 50 of the 60 cases, so the check tells the fixed-prior result apart from the uniform one.
What it does not say
- Nothing for infinitely many states or infinite outcome sets, where the minimax step needs compactness arguments not given here.
- Nothing for signed losses unless they are measured by their range: with ‖L‖∞ the bound can be off by a factor of 2 (NOTE-1430).
- No kernel form for the deficiency restricted to a family Q of problems. That is CLAIM-381’s finite computation, and it is bounded by this one.
- Not that the deficiency tells rendering procedures apart, which is CLAIM-228’s empirical question.
Connections
CLAIM-326 and CLAIM-380 state the Bayes-risk comparison for a stated prior, and each says it does not state “the prior-free form or its exact relation to Le Cam’s deficiency”. This claim states that form for finite sets: the deficiency is the least ε by which every [0, 1]-loss problem can be matched at every state. CLAIM-488 rests on it.
Note of 2026-10-31: an ex ante criterion of failing to inform
THEORY-1276 reads Simondon, through Iliadis and Bencherki & Iliadis, on information. Information is a disparity that transduction resolves. It has no “unity and identity” apart from a system (Iliadis, pp. 10–11). Its meaning is the structure the resolution produces, so it is known only afterwards (“retrospectively”, LIT-1902, p. 1357). The readers left open whether anything counts as a failure to inform (NOTE-1485). This result agrees on the first point and, in its own terms, answers the second. Here too an experiment is informative only relative to decision problems: it has no informational content apart from what it lets a receiver do. The comparison, though, comes before any outcome. The deficiency is computed from the kernels, and an experiment within ε of an uninformative one improves no decision problem with losses in [0, 1] by more than ε. So the decision-theoretic notion has what the Simondonian one lacks, an ex ante criterion of failure. The price is fixing the states and the receivers’ problems in advance, which is what the process readings decline to do: their domains are individuated “for practical purposes” (NOTE-1483). This is a translation, not an identity. Simondon’s disparity is not a likelihood, and nothing in the readings says how to write it as one. No status, title or warrant changes.