Traditional deep Gaussian processes model the data evolution using a discrete hierarchy, whereas differential Gaussian processes (DIFFGPs) represent the evolution as an infinitely deep Gaussian process. However, prior DIFFGP methods often overlook the uncertainty of kernel hyperparameters and assume them to be fixed and time-invariant, failing to leverage the unique synergy between continuous-time models and approximate inference. In this work, we propose a fully Bayesian approach that treats the kernel hyperparameters as random variables and constructs coupled stochastic differential equations (SDEs) to learn their posterior distribution and that of inducing points. By incorporating estimation uncertainty on hyperparameters, our method enhances the model's flexibility and adaptability to complex dynamics. Additionally, our approach provides a time-varying, comprehensive, and realistic posterior approximation through coupling variables using SDE methods. Experimental results demonstrate the advantages of our method over traditional approaches, showcasing its superior performance in terms of flexibility, accuracy, and other metrics. Our work opens up exciting research avenues for advancing Bayesian inference and offers a powerful modeling tool for continuous-time Gaussian processes.

翻译：传统的深度高斯过程采用离散层次结构对数据演化进行建模，而微分高斯过程（DIFFGPs）则将演化过程表示为无限深的高斯过程。然而，现有的DIFFGP方法往往忽略核超参数的不确定性，将其视为固定且时不变的参数，未能充分利用连续时间模型与近似推断之间的独特协同效应。本研究提出一种完全贝叶斯方法，将核超参数作为随机变量处理，并通过构建耦合随机微分方程（SDEs）来学习其与诱导点的后验分布。通过纳入超参数的估计不确定性，我们的方法增强了模型对复杂动态的灵活性和适应能力。此外，通过采用SDE方法对变量进行耦合，本方法能够提供时变的、全面的且符合现实的后验近似。实验结果表明，相较于传统方法，我们的方法在灵活性、准确性及其他指标上均展现出优越性能。本研究为推进贝叶斯推断开辟了新的研究路径，并为连续时间高斯过程提供了强大的建模工具。