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 this calibrator yields substantial efficiency gains over existing p-to-e methods. This $e$-value formulation enables principled use of recent advances in $e$-value merging and randomization to improve conformal inference. 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 achieve exact $1-\alpha$ coverage 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.