Either a Confidence Interval Covers, or It Doesn't (Or Does It?): A Model-Based View of Ex-Post Coverage Probability

TL;DR

This paper challenges the traditional view that confidence intervals only indicate whether a parameter is covered or not, proposing a model-based perspective that allows for probabilistic statements about individual intervals.

Contribution

It introduces a formal and informal argument against the strict dichotomy of coverage, framing confidence as a predictive probability within a unified model.

Findings

Recasts confidence intervals in terms of infinite trial sequences.

Proposes confidence as a predictive probability about coverage.

Highlights tension between behavioristic interpretation and mathematical machinery.

Abstract

In Neyman's original formulation, a 1-alpha confidence interval procedure is justified by its long-run coverage properties, and a single realized interval is to be described only by the slogan that it either covers the parameter or it does not. On this view, post-data probability statements about the coverage of an individual interval are taken to be conceptually out of bounds. In this paper, I present two kinds of arguments against treating that "either-or" reading as the only legitimate interpretation of confidence. The first is informal, via a set of thought experiments in which the same joint probability model is used to compute both forward-looking and backward-looking probabilities for occurred-but-unobserved events. The second is more formal, recasting the standard confidence-interval construction in terms of infinite sequences of trials and their associated 0/1 coverage indicators. In that representation, the design-level coverage probability 1-alpha and the degenerate conditional probabilities given the full data appear simply as different conditioning levels of the same model. I argue that a strict behavioristic reading that privileges only the latter is in tension with the very mathematical machinery used to define long-run error rates. I then sketch an alternative view of confidence as a predictive probability (or forecast) about the coverage indicator, together with a simple normative rule for when intermediate probabilities for single coverage events should be allowed. Keywords: confidence intervals; coverage probability; frequentist inference; single-case probability; predictive probability; Neyman. Disclaimer: The findings and conclusions in this report are those of the author and do not necessarily represent the official position of the Centers for Disease Control and Prevention.