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

Theorem: Visibility and log-contrast confidence intervals

From detector records to log-visibility confidence › Route A: invert calibrated event laws · lines 2150–2200 · proof included or linked

En01-B@2#thm:logCI

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At $k$ settings, use $N_{i}$ independent attempted runs and empirical event frequencies $q_{i}$ . Let

u_{i} = \frac{1}{2 N _{i}} lo g \frac{2 k}{α}, q_{i}^{-} = max (0, q_{i} - u_{i}), q_{i}^{+} = min (1, q_{i} + u_{i}) .

With probability at least $1 - α$ , all event probabilities lie in these intervals.

On the calibration coverage event, valid visibility enclosures are

V_{i}^{-} = max {V_{m i n}, \frac{q _{i}^{-} - a _{i}^{+} - ϵ _{det, i}}{κ _{i}^{+}}},

V_{i}^{+} = min {1, \frac{q _{i}^{+} - a _{i}^{-} + ϵ _{det, i}}{κ _{i}^{-}}} .

If an enclosure is empty, the response model and data are inconsistent at the declared confidence level. Otherwise

Y_{i} \in [- lo g V_{i}^{+}, - lo g V_{i}^{-}] .

The interval for $C = \sum_{i} w_{i} Y_{i}$ is obtained by selecting lower or upper endpoints according to the sign of each weight. It has simultaneous coverage at least $1 - α - β$ if the calibration event has probability at least $1 - β$ .

Proof. Apply two-sided Hoeffding to each event and a union bound. For positive $V_{i}$ , the response equation implies

q_{i} \leq a_{i}^{+} + κ_{i}^{+} V_{i} + ϵ_{det, i}, q_{i} \geq a_{i}^{-} + κ_{i}^{-} V_{i} - ϵ_{det, i} .

Rearrange using the observed probability bounds. The logarithm is monotone, and a signed linear combination of intervals has the stated endpoint rule. Combine the two coverage events by a union bound.

Displayed formulas in this passage

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$u_{i} = \frac{1}{2 N _{i}} lo g \frac{2 k}{α}, q_{i}^{-} = max (0, q_{i} - u_{i}), q_{i}^{+} = min (1, q_{i} + u_{i}) .$
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$V_{i}^{-} = max {V_{m i n}, \frac{q _{i}^{-} - a _{i}^{+} - ϵ _{det, i}}{κ _{i}^{+}}},$
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$V_{i}^{+} = min {1, \frac{q _{i}^{+} - a _{i}^{-} + ϵ _{det, i}}{κ _{i}^{-}}} .$
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$Y_{i} \in [- lo g V_{i}^{+}, - lo g V_{i}^{-}] .$
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$q_{i} \leq a_{i}^{+} + κ_{i}^{+} V_{i} + ϵ_{det, i}, q_{i} \geq a_{i}^{-} + κ_{i}^{-} V_{i} - ϵ_{det, i} .$

Theorem: Visibility and log-contrast confidence intervals

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