Theorem: Cue-conditioned classification reference

For a uniform rule prior, the Bayes accuracy with the current cue is

If the cue reveals the rule with probability $1-\rho$ and otherwise reports a recognizable erasure,

Further rule-independent processing of a cue cannot improve its Bayes accuracy.

Proof. The inner sum is the joint conditional probability of cue $c$ and label $\ell$ at a fixed point. Optimize the prediction separately for every cue.

A nonerased cue gives accuracy one. An erased cue leaves the uncued reference $3/4$. Average these two cases.

A predictor receiving a degraded cue can be simulated by a predictor receiving the original cue and applying the degrading channel before its decision.