OBCP: Online Backward Conformal Prediction under Distribution Shift

Backward conformal prediction (BCP) controls prediction-set size through a data-dependent level. In online prediction, true labels are revealed after predictions are made. As the data distribution changes, the true-label BCP ratio may no longer satisfy the e-value expectation condition, and past data may no longer reflect current risk. We propose Online Backward Conformal Prediction (OBCP), extending BCP to dynamic prediction and risk reporting under a fixed set-size budget. We introduce a conditional inflation factor and construct a clipped product surrogate to bound miscoverage risk without relying on the e-value condition. We derive a three-term risk bound combining a recency-weighted average of past clipped products, a target-drift term, and a finite-sample concentration term. When a valid drift allowance is available, this bound yields fixed-time high-probability certificates for current conditional miscoverage. We also establish exact quantitative relations between the clipped-product statistic and empirical error counting. Simulations compare oracle-drift certificates, while a chronological real-data audit evaluates the observable product statistic.