We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides calibrated prediction intervals. Unlike the 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. We find empirically that BCI avoids uninformative intervals that have infinite lengths and generates substantially shorter prediction intervals on volatility forecasting problems when compared with existing methods.

翻译：我们提出了贝尔曼共形推断（BCI）框架，该框架可集成于任意时间序列预测模型并提供已校准的预测区间。与现有方法不同，BCI能够利用多步超前预测，通过在每个时间步求解一维随机控制问题（SCP）来显式优化平均区间长度。具体而言，我们采用动态规划算法寻找SCP的最优策略。我们证明了BCI在任意分布偏移和时间依赖条件下，即便在使用较差的多步超前预测时，仍能实现长期覆盖。实证结果表明，与现有方法相比，BCI能够避免产生无限长度的无效区间，并在波动率预测问题上生成显著更短的预测区间。