Identifiability from Stationary Covariance in Linear SDEs with Reciprocal Interactions and Latent Variables
Continuous-time linear SDEs provide a natural model for interacting dynamical systems, but recovering causal drift coefficients from equilibrium data is difficult when reciprocal interactions (directed 2-cycles) or latent variables are present. This paper establishes new graphical sufficient conditions for generic identifiability of the drift matrix from stationary covariance in non-simple graphs, and develops an instrumental-variable framework to identify target direct effects under latent confounding. Both contributions extend the rank-based algebraic approach of Dettling et al. (2023) to settings previously unexplored for stationary linear SDEs.
The paper delivers a credible theoretical advance: it broadens identifiability guarantees for graphical continuous Lyapunov models beyond simple graphs and proposes a principled IV approach under partial observability. Nevertheless, the results remain purely population-level, the sufficient graphical conditions are rather restrictive in denser regimes, and no estimation algorithm or statistical analysis is offered.
The unified rank-based perspective is compelling: the authors consistently ground identifiability in nonzero minors of the kernel matrix $\mathbf{H}(\boldsymbol{\Sigma})_{E^c,\cdot}$ from Lemma 1. Theorem 1 supplies a transparent sufficient condition for vertex-disjoint 2-cycles with distinct witnesses, Proposition 1 gives exact rational certificates for four-node motifs, and Theorem 3 translates instrumental-variable logic cleanly to the Lyapunov setting. Theorem 2 on subgraph identifiability is a clean and useful localization result.
The contributions are entirely theoretical and lack any finite-sample analysis, estimator, or real-data validation; the conclusion explicitly flags statistical properties as future work. Moreover, Theorem 1 captures fewer identifiable graphs as density increases, certifying only 6.4% of cases for $|V|=12$ at edge probability $0.30$, which suggests its vertex-disjoint witness structure is far from necessary.\nThe IV framework in Theorem 3 demands strong root-instrument and trek-separation assumptions that may be hard to satisfy in practice, and Theorems 4 and 5 give intricate graphical rank criteria that are only sufficient, leaving the intermediate-vertex case without a complete characterization.
The authors fairly situate their work relative to Dettling et al. (2023) and to SEM identifiability literature, and they correctly note that their IV approach differs from prior sign-identifiability or integrated-covariance proposals. The only empirical evidence consists of a small bottleneck simulation showing recovery beyond the sufficient condition of Theorem 5 and a coverage analysis of Theorem 1; neither includes comparison with competing estimators or real systems.
The theoretical derivations are carefully documented in the appendices, so the proofs can in principle be checked by hand. However, the computational experiments rely on supplementary material and supplemental code that are referenced but not linked in the arXiv text, meaning independent reproduction would require reimplementation from the high-level protocol in Appendix D.
Continuous-time stochastic differential equations (SDEs) provide a natural framework for modeling interacting dynamical systems. A central causal question is whether causal effects among observed variables can be identified from observational data drawn from the stationary distribution, given the underlying causal graph. For linear SDEs, existing identifiability results are largely restricted to fully observed systems whose causal graphs contain no directed 2-cycles. We first establish new graphical conditions for the generic identifiability of causal effects in systems that may contain directed 2-cycles, thereby allowing for reciprocal interactions. Next, we address the previously unexplored setting of latent variables by developing an instrumental-variable framework for identifying a target direct causal effect under partial observability.
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