Set-Preserving Calibration from Conformal P-Values to E-Values

Standard conformal prediction (CP) procedures are typically formulated in terms of p-values, but reliance on p-values alone limits flexibility, for example, when combining dependent evidence across models or data splits. Recent work has explored e-value formulations for conformal inference, yet a direct connection between p- and e-value formulations in CP has been missing, especially regarding their statistical efficiency. We first identify limitations of classical p-to-e calibrators in the CP setting, showing that they are not set-preserving and can lead to overly conservative prediction sets. To address this, we propose a novel P2E calibrator that converts conformal p-values into e-values without altering the prediction set induced by the original conformal p-value. We establish both theoretically and empirically that our calibrator can yield significant efficiency gains over existing p-to-e calibrators. This e-value formulation enables principled use of recent advances in e-value merging and randomization, where we demonstrate its impact in two applications: cross-conformal prediction (CCP), whose variants typically provide only approximate $1-2\alpha$ coverage, and conformal aggregation (CA). In both cases, our e-value-based methods satisfy the desired $1-\alpha$ coverage guarantee while improving efficiency over standard baselines. More broadly, our approach expands the flexibility of CP and opens new directions for efficient, distribution-free uncertainty quantification.

Modern machine learning systems often make predictions without clearly communicating how uncertain they are. Conformal prediction addresses this by producing prediction sets that contain the true unkown answer with a user-specified probability, but most conformal methods are built around p-values, which can be hard to combine across models or data splits. This paper develops a way to convert conformal p-values into e-values while preserving the original prediction set. This means we keep the practical efficiency of standard conformal prediction while gaining access to the rich e-value theory. We apply this idea to two conformal prediction settings: cross-conformal prediction and conformal aggregation, showing that the resulting methods retain valid coverage and can produce smaller prediction sets. The broader goal is to make uncertainty quantification more flexible, reliable, and useful in applications where calibrated predictive uncertainty matters.