Coherent frequentism

TL;DR

This paper introduces a framework called coherent frequentism that uses dual probability measures to secure coherent subjective inference without priors, linking confidence sets with decision theory and logical consistency.

Contribution

It develops a novel decision-theoretic approach using dual frequentist posteriors that ensures coherence and logical consistency in inference without relying on prior distributions.

Findings

Frequentist posteriors satisfy decision-theoretic and logical axioms.

Confidence levels from these measures converge to 1 or 0 depending on hypothesis truth.

The framework unifies confidence intervals with coherent probabilistic inference.

Abstract

By representing the range of fair betting odds according to a pair of confidence set estimators, dual probability measures on parameter space called frequentist posteriors secure the coherence of subjective inference without any prior distribution. The closure of the set of expected losses corresponding to the dual frequentist posteriors constrains decisions without arbitrarily forcing optimization under all circumstances. This decision theory reduces to those that maximize expected utility when the pair of frequentist posteriors is induced by an exact or approximate confidence set estimator or when an automatic reduction rule is applied to the pair. In such cases, the resulting frequentist posterior is coherent in the sense that, as a probability distribution of the parameter of interest, it satisfies the axioms of the decision-theoretic and logic-theoretic systems typically cited in support of the Bayesian posterior. Unlike the p-value, the confidence level of an interval hypothesis derived from such a measure 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.