建模不确定性对于回归算法的可靠性和可解释性至关重要。回归算法中最基础的算法为直接回归算法，可通过将其输出映射到某一概率分布来建模其不确定性。在实际应用中，基于排序和基于分类的回归算法通常具有更好的性能。如何对上述现行的回归算法进行不确定性建模仍是一个开放问题。
本研究提出了一种有序概率嵌入方法，将样本表示为嵌入空间中的多元高斯分布，而非确定性点估计嵌入，这使得该方法可轻松融入现行的回归算法之中。针对有序回归任务的特点，即输出空间具有有序性的特点，提出了一种针对嵌入空间的有序性约束。
大量实验表明通过建模数据不确定性以及约束嵌入有序性，有序概率嵌入方法在多个任务上均达到和超过了目前最先进算法的性能。同时本方法可便捷地计算单个样本的不确定性分数，为下游任务提供额外的不确定性信息。本方法对研发高鲁棒性、可解释、风险可控的机器学习系统具有重要意义。

Modeling uncertainty is of vital importance to the reliability and explainability of regression algorithms. Direct regression is the fundamental method in the field, whose uncertainties can be modeled by mapping its output space to a certain family of probabilistic distributions. In real-world applications, classification-based or ranking-based regression is more popular, and how to model uncertainty with the above regressors is still an open problem.
This thesis proposes probabilistic ordinal embeddings, which represent each data as a multivariable Gaussian. Also, it could be incorporated into the above regression algorithms with no effort. To address the ordinal-property problem, we propose an ordinal constraint in latent embedding space.
Extensive experiments indicate that our method achieves competitive performance compared with state-of-the-art approaches. Our method can easily estimate per-exemplar uncertainty score, which can be utilized by downstream tasks. The proposed method facilitates the development of highly robust, explainable, and risk-controllable machine learning systems.