Manuscript Psy03-D@1 · Psy03-D-v1.tex · theorem · 3e18951d

Theorem: Hidden-rule observation obstruction

An explicitly specified hidden-rule task › Stationarity and the observation obstruction · lines 356–393 · proof included or linked

Psy03-D@1#thm:observability, Psy03-D@1#eq:labelposterior, Psy03-D@1#eq:generalbayes

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Under Assumption ass:uncued,

P (R_{t} = r ∣ X_{0 : T - 1}, B, ϑ) = \frac{1}{4},

where $ϑ$ denotes the fixed assessed checkpoint. Consequently,

P (Y_{t} = c ∣ X_{0 : T - 1}, B, ϑ) = \frac{1}{4} r = 0 \sum^{3} 1 {f_{r} (X_{t}) = c} .

The optimal accuracy at step $t$ is

Q_{t}^{*} = E [c \in {0, 1, 2, 3} max \frac{1}{4} r = 0 \sum^{3} 1 {f_{r} (X_{t}) = c}] .

No function of the input history can improve on this reference. Knowing switch times does not reveal the new rule identity.

Proof. Lemma lem:uniform and assessment independence give the rule posterior. The target is a deterministic function of $R_{t}, X_{t}$ , giving (eq:labelposterior). Conditional randomized classification is a convex combination of the four possible label probabilities and cannot exceed their maximum. Selecting a maximizing label attains that maximum.

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Psy03-D@1#eq:labelposterior · line 364
$P (Y_{t} = c ∣ X_{0 : T - 1}, B, ϑ) = \frac{1}{4} r = 0 \sum^{3} 1 {f_{r} (X_{t}) = c} .$
unlabeled · line 359
$P (R_{t} = r ∣ X_{0 : T - 1}, B, ϑ) = \frac{1}{4},$
Psy03-D@1#eq:generalbayes · line 371
$Q_{t}^{*} = E [c \in {0, 1, 2, 3} max \frac{1}{4} r = 0 \sum^{3} 1 {f_{r} (X_{t}) = c}] .$

Theorem: Hidden-rule observation obstruction

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