详解多输出高斯过程的推导与梯度计算，助你理解其原理。
Derivation of Output Correlation Inferences for Multi-Output (aka Multi-Task) Gaussian Process
- 通过清晰推导多任务高斯过程的数学公式和梯度
- 提供从联合分布到协方差矩阵的完整推导路径
- 适合学习贝叶斯优化或多任务建模的研究者
高斯过程（GP）是实践中最广泛使用的机器学习算法之一，尤其在贝叶斯优化（BO）中应用广泛。尽管单任务GP已具强大能力，但考虑多个输出之间的依赖关系往往能提升性能。为此，多任务高斯过程（MTGP）被提出，然而现有文献中其公式的推导及梯度计算并不直观。本文提供了对MTGP公式及其梯度的友好推导，帮助读者系统理解其数学基础。
原文摘要 · Abstract (English)
Gaussian process (GP) is arguably one of the most widely used machine learning algorithms in practice. One of its prominent applications is Bayesian optimization (BO). Although the vanilla GP itself is already a powerful tool for BO, it is often beneficial to be able to consider the dependencies of multiple outputs. To do so, Multi-task GP (MTGP) is formulated, but it is not trivial to fully understand the derivations of its formulations and their gradients from the previous literature. This paper serves friendly derivations of the MTGP formulations and their gradients.
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