Reconciliation enforces coherence between hierarchical forecasts, in order to satisfy a set of linear constraints. However, most works focus on the reconciliation of the point forecasts. We instead focus on probabilistic reconciliation and we analyze the properties of the reconciled distributions by considering reconciliation 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 behavior of 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 about the mean and variance of the reconciled distribution and show that reconciliation yields a major gain in forecasting accuracy compared to the base forecasts.

翻译：协调方法通过满足一组线性约束，确保层次预测之间的一致性。然而，现有研究大多聚焦于点预测的协调。本文转而关注概率协调，通过条件化协调方法分析协调分布的性质。我们给出了协调分布方差的正式分析，分别处理高斯预测与计数预测的情况。同时，针对单层层级结构，我们研究了协调上均值的特性，并在此情形下同样分别分析了高斯预测与计数预测的表现。随后，我们基于极端市场事件计数的间歇性时间序列数据开展协调实验。实验结果验证了关于协调分布均值与方差的理论推导，并表明相较于基础预测，协调方法能显著提升预测精度。