The relation between frequentist confidence intervals and Bayesian credible intervals

TL;DR

This paper explores the connection between frequentist confidence intervals and Bayesian credible intervals by identifying a specific prior that aligns Bayesian results with frequentist methods, often matching the Jeffreys prior.

Contribution

It derives a 'frequentist' prior that makes Bayesian and frequentist confidence intervals equivalent, clarifying their relationship and conditions for equivalence.

Findings

The 'frequentist' prior often coincides with the Jeffreys prior.

The derived prior leads to identical confidence intervals in many cases.

The relationship between the two approaches depends on the specific problem.

Abstract

We investigate the relation between frequentist and Bayesian approaches. Namely, we find the "frequentist" Bayes prior \pi_{f}(\lambda,x_{obs}) = -\frac{\int_{-\infty}^{x_{obs}}\frac{\partial f(x,\lambda)}{\partial \lambda}dx}{f(x_{obs},\lambda)} (here f(x,\lambda) is the probability density) for which the results of frequentist and Bayes approaches to the determination of confidence intervals coincide. In many cases (but not always) the "frequentist" prior which reproduces frequentist results coincides with the Jeffreys prior.