Forecast reconciliation, an ex-post technique applied to forecasts that must satisfy constraints, has been a prominent topic in the forecasting literature over the past two decades. Recently, several efforts have sought to extend reconciliation methods to the probabilistic settings. Nevertheless, formal theorems demonstrating error reduction in nonlinear constraints, analogous to those presented in Panagiotelis et al.(2021), are still lacking. This paper addresses that gap by establishing such theorems for various classes of nonlinear hypersurfaces and vector-valued functions. Specifically, we derive an exact analog of Theorem 3.1 from Panagiotelis et al.(2021) for hypersurfaces with constant-sign curvature. Additionally, we provide an error reduction theorem for the broader case of hypersurfaces with non-constant-sign curvature and for general manifolds with codimension > 1. To support reproducibility and practical adoption, we release a JAX-based Python package, JNLR, implementing the presented theorems and reconciliation procedures.

翻译：预测协调是一种应用于必须满足约束条件的预测的事后处理技术，在过去二十年中一直是预测文献中的重要课题。最近，多项研究尝试将协调方法扩展到概率化场景。然而，针对非线性约束的误差缩减形式化定理，类似于Panagiotelis等人（2021）提出的结果，目前仍然缺失。本文通过为各类非线性超曲面和向量值函数建立此类定理来填补这一空白。具体而言，我们针对具有恒定符号曲率的超曲面，推导出Panagiotelis等人（2021）中定理3.1的精确对应形式。此外，我们还为更广泛的非恒定符号曲率超曲面情形以及余维数>1的一般流形提供了误差缩减定理。为支持可复现性和实际应用，我们发布了基于JAX的Python软件包JNLR，其中实现了本文提出的定理与协调流程。