An Asymmetric Independence Model for Causal Discovery on Path Spaces

TL;DR

This work introduces E-separation, an asymmetric, time-aware separation criterion for directed mixed graphs to enable causal discovery from path-space observations in stochastic differential equations, including cyclic dynamics and partial observability. It develops a lifted dependency graph framework, defines asymmetric graphoid properties, and proves a dynamic global Markov property linking CI on path space to E-separation. The paper characterizes Markov equivalence under E-separation, proves the existence of a greatest element within each equivalence class for fully observed graphs, extends these results to latent models via latent projection, and provides experimental validation including an algorithm for causal discovery on SDEs. These results advance causal inference in continuous-time dynamical systems by leveraging time directionality and providing provable, data-recoverable structure in the presence of cycles and hidden variables.

Abstract

We develop the theory linking 'E-separation' in directed mixed graphs (DMGs) with conditional independence relations among coordinate processes in stochastic differential equations (SDEs), where causal relationships are determined by "which variables enter the governing equation of which other variables". We prove a global Markov property for cyclic SDEs, which naturally extends to partially observed cyclic SDEs, because our asymmetric independence model is closed under marginalization. We then characterize the class of graphs that encode the same set of independence relations, yielding a result analogous to the seminal 'same skeleton and v-structures' result for directed acyclic graphs (DAGs). In the fully observed case, we show that each such equivalence class of graphs has a greatest element as a parsimonious representation and develop algorithms to identify this greatest element from data. We conjecture that a greatest element also exists under partial observations, which we verify computationally for graphs with up to four nodes.

Figures (6)

• Figure 1: The lifted dependency graph $\tilde{ \mathcal{G}}$ for the DMG $\mathcal{G}$.
• Figure 2: DGs (i)-(xii) are the elements of one Markov equivalence and (xii) is its greatest element.
• Figure 3: An inducing path connects $X^1$ and $X^3$ in $\mathcal{G}$, and $X^3$ (red) E-separates $X^3$ (green) from $X^1$ (teal).
• Figure 4: Causal discovery for SDEs.
• Figure 5: Results from applying the algorithm to two 3-dimensional linear SDEs, showing predicted probabilities for each edge and confirming the expected equivalence classes.

Theorems & Definitions (29)

• Definition 2.1: Directed (Mixed) Graph, D(M)G
• Proposition 3.1: Ternary relation defined by $\mathop{\mathrm{\perp\!\!\!\perp}}\nolimits_{E}^{ \mathcal{G}}$
• Definition 3.2: Separability
• Lemma 3.3
• Definition 3.4: Future-extended $h$-locally CI
• Remark
• Proposition 3.5: $\mathop{\mathrm{\perp\!\!\!\perp}}\nolimits_{s,h}^+$ as a ternary relation
• Proposition 3.6: Global Markov property for E-separation and $\mathop{\mathrm{\perp\!\!\!\perp}}\nolimits_{s,h}^+$
• Definition 3.7: Markov equivalence
• Definition 3.8: Maximal DMGs