The case
A stipulated toy. An organization’s processes i = 1, …, k hold authority shares π on the simplex. Under a probe context c, process i would respond with pᵢ(y | c), and the organization’s response is the mixture p_π(y | c) = Σ πᵢ pᵢ(y | c). That is CLAIM-444’s mixture condition. The profiles pᵢ are known; only π is to be learned.
- Authority geometry. If the evaluator saw which process answered, the Fisher metric on π would be the categorical one, g(v, w) = Σ vᵢwᵢ/πᵢ on tangent vectors (Σ vᵢ = 0). That is the Shahshahani metric, in which the replicator ascends a potential with dΦ/dt = Var_π(q) (CLAIM-444, CASE-155 part 3).
- Response geometry. Seeing only y, the metric is the pullback G_resp = Jᵀ diag(p_π)⁻¹ J, where J = ∂p_π/∂π along the simplex. Its columns are differences of the processes’ profiles.
- Never more. The response y is the process index passed through a channel that does not depend on π, so G_resp ⪯ G_auth. That is CLAIM-450’s garbling inequality, applied to π.
- Redundancy. If p₁(· | c) = p₂(· | c) for every probe used, then Jv = 0 for v = e₁ − e₂, so g_auth(v, v) > 0 while G_resp(v, v) = 0. With π = (0.3, 0.7) the two values are 4.76 and 0.
- A rare context. Two specialists both answer yes with probability 0.8 in routine contexts, and with probabilities 0.9 and 0.1 in a rare one. With π the first specialist’s share, the routine response stays at 0.8, so I_routine(π) = 0. The rare response is p = 0.1 + 0.8π, so I_rare(π) = 0.64/(p(1 − p)): 2.56 at π = ½ and 3.33 at π = 0.2 or 0.8, against the authority metric’s 1/(π(1 − π)) of 4 and 6.25. A31: “No number of routine observations can identify π from these responses alone. Rare-context observations can.”
Checked when filed, by script (numpy): the pullback against the Fisher information computed directly by finite differences of the log likelihood; zero along the redundant direction; the smallest eigenvalue of G_auth − G_resp on the tangent space nonnegative on 200 random three-process, four-response mixtures; and the rare-context values above.
Three redundancies
A29’s table, kept because the record’s distinctions need it: “Parameter redundancy | Different parameter settings of a fitted model”; “Observational redundancy | Different organizations under the current probe family”; “Functional redundancy | Organizations under all interventions relevant to a declared task”. “These need not coincide.” This case is the second without the first, since the profiles are known and the model is regular in π. It says nothing about the third.
Choosing the probes
Choosing a design ξ over contexts to maximize a criterion of G_ξ = Σ_c ξ(c)G_c is a nonlinear design problem in Chaloner and Verdinelli’s sense (NOTE-1161), because the information depends on π. An optimal design has to be stated at a guess of π or averaged over a prior on it, and local optimality is the point-mass case (THEORY-954). Adaptive probing, which updates the guess between probes, should gain over a fixed design there, as it does not in the normal linear model (THEORY-956). In the two-specialist case above the answer does not depend on π: the routine context carries no information, so every criterion that increases with information puts all mass on the rare one.
A three-process illustration, checked by script. Four contexts with response profiles (rows contexts, columns processes) (0.8, 0.8, 0.8), (0.9, 0.1, 0.5), (0.6, 0.6, 0.1) and (0.7, 0.75, 0.72), at π = (0.5, 0.3, 0.2). The D-optimal design is ξ = (0, ½, ½, 0), all weight on the two contexts that separate the processes. Its sensitivities tr(G_ξ⁻¹G_c) are (0, 2, 2, 0.0097), none above the dimension 2 of the tangent space, which certifies it by the equivalence theorem. Its log det is −0.42, against −1.80 for the uniform design and −5.02 for a routine-heavy one (0.85, 0.05, 0.05, 0.05). The values hold at this π only.
What it can show
That an organization can reorganize internally while its judgements stay the same under the probes in use. That the remedy is a different probe, not a different fit (CLAIM-305). A29: “If the experts differ on a rare class of questions, the experiment should ask those questions. If they differ only when given different authority, we need an authority intervention.” And that the geometry in which authority moves is not the geometry in which evidence about it arrives.
What it cannot show
- Anything when the profiles pᵢ are themselves unknown. The model is then singular where they coincide, and CASE-152’s analysis applies, not this one. A31: “we’d have to analyze the fitted likelihood rather than assume singularity implies a particular learning coefficient.”
- That choosing probes well removes the problem. Probes on which the processes disagree must also be accessible to the evaluator, which is QUESTION-271’s failure of bandit feedback. An inaccessible probe is an action outside the set D of THEORY-962, and what the remaining probes cannot see stays unlearnable however long they are used.
- That two processes that agree under the probes are the same communicative kind, or differ as one (CLAIM-150).