AI 中文总结
研究随机微分方程的图形因果模型，提出因果SDE系统可解性条件，建立基于因果图的马尔可夫性质和do-演算，证明一类SDEs的更强性质，引入时间分割系统，探讨因果发现算法在框架内对SDEs的应用。
AI 中文摘要
随机微分方程（SDEs）广泛用于对连续时间动力系统建模，但其图形因果模型尚未被充分理解。本文考虑具有明确因果语义的因果SDE系统，提出其可解性条件，使其具有明确定义的观测和干预分布，给出满足条件的Lipschitz半鞅SDEs的一般类。建立了基于系统因果图的σ-分离马尔可夫性质和do-演算，证明了一类加性噪声SDEs更强的d-分离马尔可夫性质。引入时间分割系统用于推理子采样时间序列等。最后讨论了基于约束的因果发现算法在框架内对SDEs的应用。
英文摘要
Stochastic differential equations (SDEs) are widely used to model continuous-time dynamical systems, but graphical causal models for them are not yet well-understood. We consider systems of causal SDEs that are equipped with an explicit causal semantics. We pose solvability conditions for systems of causal SDEs such that they have well-defined observational and interventional distributions - even after marginalisation - and provide a general class of Lipschitz semimartingale SDEs that satisfies these conditions. As core results we establish the $σ$-separation Markov property and the do-calculus in terms of the system's causal graph for probabilistic independence and interventions on the level of sample paths. For a class of additive-noise SDEs we prove a stronger $d$-separation Markov property, even if the system is cyclic. As a corollary of the do-calculus, we obtain an explicit causal interpretation of the graph: that the absence of a directed path implies the absence of a causal effect. We further introduce time-split systems, which consider the causal relations between the processes when evaluated on disjoint intervals or time-points, and use them to reason about subsampled time-series, continuous-time Granger non-causality and local independence. Finally, we discuss how constraint-based causal discovery algorithms (PC, FCI, CCD, CCI) apply directly to SDEs within our framework when conditional independence between sample paths can be consistently tested.
