The relation between frequentist confidence intervals and Bayesian credible intervals

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.

💡 Research Summary

The paper investigates the relationship between frequentist confidence intervals and Bayesian credible intervals, focusing on the conditions under which the two approaches yield identical interval estimates for a parameter λ. Starting from the Bayesian posterior density
  p(λ|x_obs) = f(x_obs, λ) π(λ) / ∫ f(x_obs, λ′) π(λ′) dλ′,
and the Neyman construction for frequentist confidence belts, the authors require that the cumulative posterior probability over a candidate interval