Identifiability from Stationary Covariance in Linear SDEs with Reciprocal Interactions and Latent Variables

Ana Leticia Garcez Vicente; Saber Salehkaleybar

cs.LGOct 22, 2026

Identifiability from Stationary Covariance in Linear SDEs with Reciprocal Interactions and Latent Variables

Authors: Ana Leticia Garcez Vicente, Saber Salehkaleybar

Organizations: Leiden Institute of Advanced Computer Science (LIACS), Leiden University, Netherlands

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Generically identifiable $3$ -node motifs.

Panel of Figure 2 — Figure 2: Three-node non-identifiable motifs.

Figure 3: Recovery error of the target drift coefficients as a function of the bottleneck ratio ${|Q\rvert}/{|P^{\prime}\rvert}$ . $Q$ is the intermediate nodes and $P^{\prime}=P\setminus\{y\}$ . Curves correspond to different numbers of parents. Latent parents $\lvert U\rvert=5$ .

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Four-node motifs that are generically identifiable. Red arrows indicate admissible extensions involving the added vertex $v$ . A solid red arrow is required, while a dashed red arrow denotes an optional choice. Blue lines indicate adjacency whose orientation is arbitrary. For a set of blue adjacencies, at least one of the indicated blue connections must be present, in either direction.

Panel of Figure 5 — Figure 5: Four-node extensions that remain non-identifiable.

Figure 7: Coverage of the sufficient graphical condition in Theorem 1. Each cell reports the percentage of randomly generated non-simple graphs with numerical full kernel rank that are certified by Theorem 1, for different numbers of vertices $|V|$ and edge probabilities. .

Causal Identifiability

Causal identifiability asks when data and stated assumptions uniquely determine causal structures, effects, or counterfactuals.

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Sign Identifiability of Causal Effects in Stationary Stochastic Dynamical Systems

Gijs van Seeventer, Saber Salehkaleybar

1LIACS, Leiden University, Netherlands

Stochastic Differential Equations Structural Causal Models Causal Identifiability

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We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

Abstract

Causal Identifiability

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