An Asymmetric Independence Model for Causal Discovery on Path Spaces
This work addresses causal inference in complex dynamic systems like SDEs, which is incremental as it extends existing graphical model theory to cyclic and partially observed settings.
The paper tackles the problem of causal discovery in stochastic differential equations (SDEs) by linking E-separation in directed mixed graphs to conditional independence relations, proving a global Markov property for cyclic SDEs and developing algorithms to identify a parsimonious graph representation from data, with computational verification for graphs up to four nodes under partial observations.
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.