Confidence sets and their coverage level

Let $(S,\mathcal{S})$ be a measurable space, the data space (Measurable spaces and measurable sets), let $\mathrm{\Theta }$ be a nonempty set, the parameter set, and let $({\mathbb{P}}_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ be a family of probability measures on $(S,\mathcal{S})$ (Probability measures and probability spaces). Write $2^{\mathrm{\Theta }}$ for the power set of $\mathrm{\Theta }$. A confidence set for $\mathrm{\Theta }$ is a map $C:S⟶2^{\mathrm{\Theta }},s⟼C(s),$ such that for every $\theta \in \mathrm{\Theta }$ the set $\{s\in S:\theta \in C(s)\}$ is an element of $\mathcal{S}$. The coverage of $C$ at $\theta$ is $\gamma (\theta ):={\mathbb{P}}_{\theta }(\theta \in C),$ and, for $\alpha \in [0,1]$, $C$ has coverage level $1-\alpha$ when ${\inf }_{\theta \in \mathrm{\Theta }}\gamma (\theta )\ge 1-\alpha \mathrm{.}$

Well-definedness and conventions

The measurability requirement is exactly what makes each $\gamma (\theta )$ defined, since $\{\theta \in C\}$ is then an event in $\mathcal{S}$. The family $\{\gamma (\theta ):\theta \in \mathrm{\Theta }\}$ is nonempty and bounded below by $0$, so its infimum exists in $\mathbb{R}$ by Every nonempty set bounded below has an infimum, is unique by Suprema and infima are unique, and equals ${\inf }_{\theta }\gamma (\theta )$ as written (Greatest lower bound (infimum), Complete ordered field (least-upper-bound property)).

The level is a property of the family $({\mathbb{P}}_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ and of the map $C$; it is an infimum of probabilities of the events $\theta \in C$, calculated before the datum is observed. It contains no distribution on $\mathrm{\Theta }$ and no conditional probability given the datum; a data-dependent reading requires further structure that is not part of this definition.