Manuscript Psy04-A@1 · Psy04-A-v1.tex · theorem · d5800e2a

Theorem: Exact complete-record score

Complete attempted-run records › A specified independent-loss detector · lines 913–940 · proof included or linked

Psy04-A@1#thm:detector, Psy04-A@1#eq:recordscore

For the code and storage of Theorem thm:delay, under Assumption ass:detector,

Q_{score} (T) = \frac{1}{2} + \frac{p v ( 1 - 2 ϵ ) r _{T}}{2 2} .

The ideal complete score exceeds $3/4$ exactly when

p v (1 - 2 ϵ) r_{T} > \frac{1}{2} .

Proof. Conditional on a valid observed outcome, flipping gives

(1 - ϵ) Q_{Q} (T) + ϵ (1 - Q_{Q} (T)) = \frac{1}{2} + (1 - 2 ϵ) (Q_{Q} (T) - 1/2) .

Valid observed outcomes have probability $p v$ . All remaining outcomes contribute $1/2$ . Substitute (eq:delayscore).

Displayed formulas in this passage

Psy04-A@1#eq:recordscore · line 917
$Q_{score} (T) = \frac{1}{2} + \frac{p v ( 1 - 2 ϵ ) r _{T}}{2 2} .$
unlabeled · line 924
$p v (1 - 2 ϵ) r_{T} > \frac{1}{2} .$
unlabeled · line 931
$(1 - ϵ) Q_{Q} (T) + ϵ (1 - Q_{Q} (T)) = \frac{1}{2} + (1 - 2 ϵ) (Q_{Q} (T) - 1/2) .$