Approximate Bayesianity of frequentist confidence intervals for a binomial proportion

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion <i>p</i> are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for <i>p</i> approximates a Bayesian credible interval based on Kerman's neutral noninformative conjugate prior up to <i>O</i>(<i>n</i>⁻¹) in the confidence bounds. For the significance level α ≲ 0.317, the Bayesian interval based on the Jeffreys prior is then shown to be a compromise between the likelihood ratio and Wilson intervals.