We introduce Bellman Conformal Inference (BCI), a framework that wraps around any time series forecasting models and provides approximately 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. We find empirically that BCI avoids uninformative intervals that have infinite lengths and generates substantially shorter prediction intervals in multiple applications when compared with existing methods.

翻译：我们提出贝尔曼共形推断（BCI），这是一种可嵌入任意时间序列预测模型，并提供近似校准预测区间的框架。与现有方法不同，BCI能够利用多步超前预测，并通过在每个时间步求解一维随机控制问题（SCP）显式优化平均区间长度。具体而言，我们使用动态规划算法来寻找SCP的最优策略。我们证明，即使在多步超前预测效果不佳的情况下，BCI仍能在任意分布偏移和时间依赖性下实现长期覆盖。实验表明，与现有方法相比，BCI能够避免产生具有无限长度的非信息性区间，并在多个应用场景中生成显著更短的预测区间。