Coherent frequentism

The certainty distribution, the confidence distribution of a scalar interest parameter evaluated at fixed data and extended to a Borel space, combines the self-consistency of the Bayesian posterior distribution with the reliability of Neyman-Pearson methods. As a probability measure of the parameter, the certainty distribution is coherent in the sense that it satisfies the axioms of the decision-theoretic and logic-theoretic systems typically cited in support of the Bayesian approach. In contrast with the p-value, the certainty level of an interval hypothesis is suitable as an estimator of the indicator of hypothesis truth since it converges in sample-space probability to 1 if the hypothesis is true or to 0 otherwise under general conditions. The equality between certainty levels and the coverage rates of the corresponding confidence intervals ensures that the estimator's decision rule is uniquely minimax in a betting game designed to quantify the reliability of probability statements.