Bellman conformal inference: Calibrating prediction intervals for time series

We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting model and provides calibrated prediction intervals. Unlike existing methods, BCI is able to leverage multi-step ahead forecasts and explicitly optimize the average interval lengths by solving a one-dimensional stochastic control problem (SCP) at each time step. In particular, we use the dynamic programming algorithm to find the optimal policy for the SCP. We prove that BCI achieves long-term coverage under arbitrary distribution shifts and temporal dependence, even with poor multi-step ahead forecasts. In addition, we prove that BCI is near optimal for its surrogate model predictive control problem when the raw forecast intervals are approximately calibrated. We find empirically that BCI avoids uninformative intervals with infinite lengths and generates substantially shorter prediction intervals when the raw forecast intervals are poorly calibrated.