将高斯过程与扩散模型关联，实现任意条件下的高效概率推断。
Conditioning Gaussian Processes on Almost Anything
- 通过微分方程重述预测采样，引入可蒙特卡洛近似的引导项。
- 在非线性物理和自然语言条件下仍能准确推断，首次实现语言条件化。
- 无需定制推导，适用于真实世界复杂知识的建模，适合概率建模研究者。
高斯过程（GPs）提供了函数上的原则性概率建模，但精确推断仅限于线性-高斯情形。我们建立了高斯过程与一类线性扩散模型之间的显式等价关系，将预测采样重构为具有闭式高斯动力学的常微分方程，并引入依赖似然的引导项，该引导项可通过简单蒙特卡洛近似实现。在线性-高斯设定下，我们精确恢复标准高斯过程的条件化；在共轭之外，该方法可处理任何可进行逐点似然评估的条件声明——包括非线性物理系统，以及首次实现的通过大型语言模型进行的自然语言条件化。去相关操作分离出不可约的非高斯动态，最小化Wasserstein-2传输成本并消除数值刚性。结果是一个无需专门推导的通用高斯过程推断方案。这些成果共同提供了一种通用机制，可融入现实世界知识作为条件信息，为真实世界问题的概率建模开辟了新前沿。
原文摘要 · Abstract (English)
Gaussian processes (GPs) offer a principled probabilistic model over functions, but exact inference is restricted to the linear-Gaussian regime. We establish an explicit equivalence between GPs and a class of linear diffusion models, recasting predictive sampling as an ODE with closed-form Gaussian dynamics and a likelihood-dependent guidance term that admits a simple Monte Carlo approximation. In the linear-Gaussian setting, we recover standard GP conditioning exactly; beyond conjugacy, the same machinery handles any conditioning statement admitting point-wise likelihood evaluation -- including non-linear physics, and, for the first time, natural language via large language models. Whitening isolates the irreducible non-Gaussian dynamics, minimising Wasserstein-2 transport cost and eliminating numerical stiffness. The result is a general-purpose GP inference scheme requiring no bespoke derivations. Together, these results provide a general mechanism for incorporating the full richness of real-world knowledge as conditioning information, opening a new frontier for the probabilistic modelling of real-world problems.
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