CLs ratio for source-specified signal exclusions

Let $(S,\mathcal{S})$ be a data space with two supplied probability laws ${\mathbb{P}}_b$ and ${\mathbb{P}}_{s+b}$ for the background-only and signal-plus-background hypotheses. Fix a statistic $T:S\to \mathbb{R}$ for which larger values are more incompatible with the signal-plus-background hypothesis and for which every upper-tail set is measurable. For observed datum $x\in S$, set $R_x:=\{y\in S:T(y)\ge T(x)\}$ and define ${\mathrm{C}\mathrm{L}}_b(x):={\mathbb{P}}_b(R_x),{\mathrm{C}\mathrm{L}}_{s+b}(x):={\mathbb{P}}_{s+b}(R_x)\mathrm{.}$ When ${\mathrm{C}\mathrm{L}}_b(x)>0$, the modified-frequentist exclusion ratio is ${\mathrm{C}\mathrm{L}}_s(x):=\frac{{\mathrm{C}\mathrm{L}}_{s+b}(x)}{{\mathrm{C}\mathrm{L}}_b(x)}\mathrm{.}$ A reported exclusion rule at threshold $\alpha \in (0,1)$ excludes a specified signal-plus-background hypothesis when its supplied analysis has ${\mathrm{C}\mathrm{L}}_s(x)<\alpha$. The likelihood ratio, test statistic, nuisance treatment, and sampling laws are part of the specified analysis. The ratio is not itself a probability or a frequentist coverage level; in the cited method it is a conservative, frequentist-motivated approximation used to present exclusion results. It does not give the probability that a signal model is true or false.

Calibrated null-tail probabilities

Read, full paper, §2 and §§2.1–2.2