🤖 AI Summary
This study addresses the challenge of constructing statistical intervals that simultaneously admit a Bayesian interpretation and satisfy frequentist finite-sample coverage guarantees, all within a fully nonparametric setting. To bridge the gap between Bayesian and frequentist paradigms, the work proposes a relaxed notion of Bayesian credible intervals: rather than requiring a fixed posterior probability ex ante, it only demands that, after observing the interval, the posterior confidence be at least $p\%$. Methodologically, the approach introduces a one-dimensional prior to circumvent the complexities of high-dimensional modeling and leverages a decision-theoretic framework combined with nonparametric estimation techniques to construct intervals for both cumulative distribution functions and means of distributions with bounded support. The resulting intervals are asymptotically equivalent to those from full Bayesian procedures or slightly wider, yet they offer stronger finite-sample coverage assurances.
📝 Abstract
We propose a new type of statistical interval obtained by weakening the definition of a p% credible interval: Having observed the interval (rather than the full dataset) we should put at least a p% belief in it. From a decision-theoretical point of view the resulting intervals occupy a middle ground between frequentist and fully Bayesian statistical intervals, both practically and philosophically: To a p% Bayesian credible interval we should assign (at least a) p% belief also after seeing the full dataset, while p% frequentist intervals we in general only assign a p% belief before seeing either the data or the interval. We derive concrete implementations for two cases: estimation of the fraction of a distribution that falls below a certain value (i.e., the CDF), and of the mean of a distribution with bounded support. Even though the problems are fully non parametric, these methods require only one-dimensional priors. They share many of the practical advantages of Bayesian methods while avoiding the complexity of assigning high-dimensional priors altogether. Asymptotically they give intervals equivalent to the fully Bayesian approach and somewhat wider intervals, respectively. We discuss promising directions where the proposed type of interval may provide significant advantages.