Manuscript En01-B@2 · En01-B-v2.tex · corollary · f226d1eb

Corollary: A transparent sufficient log-contrast budget

From detector records to log-visibility confidence › Route A: invert calibrated event laws · lines 2206–2236 · proof included or linked

En01-B@2#cor:logbudget

Back to manuscript

Suppose $a_{i}, κ_{i}$ are known, with $κ_{i} \geq κ_{m i n} > 0$ , and all systematic errors have already been included in the deterministic contrast gap $m_{C} > 0$ . Estimate $V_{i}$ by inverting the event law and clipping to $[V_{m i n}, 1]$ . With equal $N$ per setting, the empirical contrast error is at most

\frac{\sum _{i} ∣ w _{i} ∣}{κ _{m i n} V _{m i n}} \frac{1}{2 N} lo g \frac{2 k}{α}

with probability at least $1 - α$ . For the three-setting weights, $\sum ∣ w_{i} ∣ = 16$ . Thus

N \geq \frac{2 ( 16 ) ^{2}}{κ _{m i n} ^{2} V _{m i n} ^{2} m _{C} ^{2}} lo g \frac{6}{α}

is sufficient to keep this error at most $m_{C} /2$ . The total attempted-run count is $3 N$ .

Proof. Clipping is nonexpansive when the true visibility belongs to the clipped interval. The inverse response has Lipschitz constant $1/ κ_{m i n}$ . On $[V_{m i n}, 1]$ , the log has Lipschitz constant $1/ V_{m i n}$ . Sum the three errors and solve the resulting inequality.

Displayed formulas in this passage

unlabeled · line 2214
$\frac{\sum _{i} ∣ w _{i} ∣}{κ _{m i n} V _{m i n}} \frac{1}{2 N} lo g \frac{2 k}{α}$
unlabeled · line 2221
$N \geq \frac{2 ( 16 ) ^{2}}{κ _{m i n} ^{2} V _{m i n} ^{2} m _{C} ^{2}} lo g \frac{6}{α}$

Corollary: A transparent sufficient log-contrast budget

Displayed formulas in this passage

Nearby in this section