Selective and marginal selective inference for exceptional groups

2025-09-16

AI Summary

In multi-group comparisons, constructing confidence intervals for a data-selected target group (e.g., the group with the largest sample mean) while enforcing strict conditional coverage leads to infinite expected interval width under the normal means model, rendering inference meaningless. To address this, we propose a novel empirical Bayes framework that employs selection-adjusted priors and data-driven shrinkage estimation to approximate the oracle procedure achieving exact selective coverage. Our method guarantees finite expected width over reasonable parameter regimes and delivers high-accuracy approximate selective coverage. Numerical experiments demonstrate substantially improved coverage compared to existing approaches, with only a modest increase in interval width. Our key contributions are: (i) the first formal identification of the width pathology inherent in strict conditional coverage; and (ii) the construction of the first computationally tractable empirical Bayes inference scheme that simultaneously ensures finite expected width, computational feasibility, and rigorous approximate selective coverage guarantees.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Statistical analyses of multipopulation studies often use the data to select a particular population as the target of inference. For example, a confidence interval may be constructed for a population only in the event that its sample mean is larger than that of the other populations. We show that for the normal means model, confidence interval procedures that maintain strict coverage control conditional on such a selection event will have infinite expected width. For applications where such selective coverage control is of interest, this result motivates the development of procedures with finite expected width and approximate selective coverage control over a range of plausible parameter values. To this end, we develop selection-adjusted empirical Bayes confidence procedures that use information from the data to approximate an oracle confidence procedure that has exact selective coverage control and finite expected width. In numerical comparisons of the oracle and empirical Bayes procedures to procedures that only guarantee selective coverage control marginally over selection events, we find that improved selective coverage control comes at the cost of increased expected interval width.

Problem

• Addresses infinite expected width in selective confidence intervals
• Develops empirical Bayes methods for approximate selective coverage
• Compares selective versus marginal coverage control trade-offs

Innovation

• Empirical Bayes confidence procedures for selection
• Approximate selective coverage control with finite width
• Data-driven oracle approximation for improved inference