Reconciliation enforces coherence between hierarchical forecasts, in order to satisfy a set of linear constraints. While most works focus on the reconciliation of the point forecasts, we consider probabilistic reconciliation and we analyze the properties of the distributions reconciled via conditioning. We provide a formal analysis of the variance of the reconciled distribution, treating separately the case of Gaussian forecasts and count forecasts. We also study the reconciled upper mean in the case of 1-level hierarchies; also in this case we analyze separately the case of Gaussian forecasts and count forecasts. We then show experiments on the reconciliation of intermittent time series related to the count of extreme market events. The experiments confirm our theoretical results and show that reconciliation largely improves the performance of probabilistic forecasting.

翻译：调和通过在层次预测之间强制一致性，以满足一组线性约束条件。尽管多数研究聚焦于点预测的调和，本文则考虑概率性调和，并分析通过条件化得到的调和分布的性质。我们分别针对高斯预测和计数预测的情况，对调和分布的方差进行了形式化分析。此外，我们在单层层次结构下研究了调和上均值，同样分别分析了高斯预测和计数预测的情况。随后，我们针对极端市场事件计数的间歇性时间序列调和开展了实验。实验结果验证了我们的理论推导，并表明调和显著提升了概率预测的性能。