AI 中文总结
本文证明预测协调中分布、汇总统计量与决策约束不可互换，刻画线性协调保留分位数的条件，并在旅游数据上展示分位数不一致及损失函数对排名的影响。
AI 中文摘要
在不同聚合层级上做出的预测通常需要保持一致——例如，区域预测应加总等于全国总数。预测协调（forecast reconciliation）施加了此类关系，但一致性的含义取决于约束是应用于预测分布、报告的汇总统计量（如均值或分位数），还是基于预测的决策。我们证明这些操作不可互换。即使预测分布的每个可能结果都满足聚合规则，其单独报告的边际分位数也可能无法相加；在非线性关系下，即使是坐标均值也可能违反约束。我们刻画了线性协调何时能保留边际分位数，并表明在真正的层级结构中，非中位数分位数通常无法被保留。我们还解释了为何在精确非线性约束条件下进行条件化需要指定该约束是如何被观测或近似的，并识别了预测均值何时包含足够信息以用于受约束的决策。在澳大利亚旅游数据（28个滚动预测起点）的应用中，在自下而上协调下，国家第95百分位数的两种自然构造之间的中位数差异为协调后90%预测区间宽度的159.9%。当平方误差损失被非对称库存损失替代时，基于均值和基于分位数的预测排名也会反转。这些结果为决定应使哪些内容保持一致以及如何评估竞争方法提供了框架。
英文摘要
Forecasts made at different levels of aggregation are often required to agree---for example, regional forecasts should sum to the national total. Forecast reconciliation imposes such relationships, but the meaning of agreement depends on whether the constraint is applied to a predictive distribution, a reported summary such as a mean or quantile, or a decision based on the forecast. We show that these operations are not interchangeable. Even when every possible outcome from a predictive distribution satisfies an aggregation rule, its separately reported marginal quantiles need not add up; under nonlinear relationships, even the coordinatewise means may violate the constraint. We characterise when linear reconciliation preserves marginal quantiles and show that nonmedian quantiles generally cannot be preserved in a genuine hierarchy. We also explain why conditioning on an exact nonlinear constraint requires specifying how that constraint is observed or approximated, and identify when the predictive mean contains enough information for a constrained decision. In an application to Australian tourism data with 28 rolling forecast origins, the median discrepancy between two natural constructions of the national 95th percentile is 159.9\% of the reconciled 90\% prediction-interval width under bottom-up reconciliation. Rankings of mean- and quantile-based forecasts also reverse when squared-error loss is replaced by asymmetric inventory loss. These results provide a framework for deciding what should be made coherent and how competing methods should be evaluated.
Comments23 pages, 5 figures, 3 tables. Replication code: https://github.com/Eurekacoding/coherence-is-not-enough
