Fisher's disjunction as the principle vindicating p-values, confidence intervals, and their generalizations: A frequentist semantics for possibility theory

Statistical evidence and surprise unified under possibility theory

“…Different rationales for the next equation may be found in coherence axioms (Bickel & Patriota, 2019) and in an argument building on Fisher's disjunction (Bickel, 2023). The compatibility of the hypothesis that $$$ \theta $$$ is in $$$ \mathscr{H} $$$ with the data is the C ‐value $$$ C\left(\mathscr{H}\right)={\sup}_{\theta_0\in \mathscr{H}}p\left({\theta}_0\right), $$$ which, in the above cases, is $$$ C\left(\left\{{\theta}_{H_0}\right\}\right)=p\left({\theta}_{H_0}\right) $$$, meaning the compatibility of the hypothesis that $$$ \theta ={\theta}_{H_0} $$$ with the data is equal to the p ‐value testing that simple hypothesis. $$$ C\left(\overline{\left\{{\theta}_{H_0}\right\}}\right)={\sup}_{\theta_0\ne {\theta}_{H_0}}p\left({\theta}_0\right)=p\left(\hat{\theta}\right)=1 $$$, meaning the compatibility of the hypothesis that $$$ \theta \ne {\theta}_{H_0} $$$ with the data is equal to 100%. $$…”

“…Restricted parameter problems, such as those with a bounded parameter of interest and those with a finite parameter space, motivated defining $$$ \mathrm{C}\left(\mathscr{H}|\mathcal{R}\right) $$$, the conditional C ‐value given that the parameter of interest is restricted to $$$ \mathcal{R} $$$, a subset of $$$ \Theta $$$ (Bickel & Patriota, 2019). Recent applications of the conditional C ‐value include interpreting initial studies in light of replication studies, modifying p ‐values for prior information, and comparing a finite number of scientific hypotheses (Bickel, 2023). The C ‐value has much in common with the measure of evidence introduced by Patriota (2013), which was recently applied to mixed models (Patriota & de Oliveira Alves, 2022).…”

“…Different rationales for the next equation may be found in coherence axioms (Bickel & Patriota, 2019) and in an argument building on Fisher's disjunction (Bickel, 2023). The compatibility of the hypothesis that 𝜃 is in  with the data is the C-value C () = sup 𝜃 0 ∈ p (𝜃 0 ) , which, in the above cases, is…”