The Exact Ville Identity: From the Absorbing Case to the General Law with an Application to E-Values

What the math gives to ML

The paper replaces Ville's inequality with an exact decomposition of sequential threshold-crossing probability. The gap is attributed to three nonnegative quantities: expected overshoot, cumulative supermartingale drift loss, and residual mass on paths that never cross. This machinery can transfer to anytime-valid monitoring of adaptively selected neural-network checkpoints, where an e-process detects improvement, regressions, or safety violations without fixing the stopping time in advance. The practical opportunity is to estimate these correction terms on independent validation streams and test whether conservative threshold recalibration reduces evaluation cost while preserving the nominal false-alarm rate.