Towards Identifiability of Interventional Stochastic Differential Equations
Aaron Zweig, Zaikang Lin, Elham Azizi, David Knowles
TL;DR
This paper investigates the conditions under which the parameters of stochastic differential equations (SDEs) can be uniquely identified from data, providing theoretical bounds and experimental validation for both linear and nonlinear cases.
Contribution
It offers the first provable bounds for SDE parameter identifiability under interventions, including tight bounds for linear SDEs and bounds for nonlinear SDEs in low noise regimes.
Findings
Provable bounds for SDE parameter recovery from stationary distributions.
Experimental validation of parameter recovery on synthetic data.
Demonstration of learnable activation functions improving gene regulatory models.
Abstract
We study identifiability of stochastic differential equations (SDE) under multiple interventions. Our results give the first provable bounds for unique recovery of SDE parameters given samples from their stationary distributions. We give tight bounds on the number of necessary interventions for linear SDEs, and upper bounds for nonlinear SDEs in the small noise regime. We experimentally validate the recovery of true parameters in synthetic data, and motivated by our theoretical results, demonstrate the advantage of parameterizations with learnable activation functions in application to gene regulatory dynamics.
Peer Reviews
Decision·Submitted to ICLR 2026
In many systems that are modeled by an SDE is unrealistic to assume that we can observe trajectories, and it is instead more common to have access to the stationary distribution. The stationary measure does not uniquely identify the drift and hence the plan of studying this problem under interventions is well motivated and interesting. The presentation is clear, and the results are easy to understand.
Establishing identifiability is only the first step towards understanding estimation accuracy; optimal procedures; computational complexity and so on. The identifiability result in the linear model appears a relatively direct fact of linear algebra. As for the nonlinear case, the result is purely perturbative and non-quantitative. It requires the drift to be a contraction with a unique fzero, enabling perturbative argument. No quantitative estimate is given on how small \epsilon must be for the
1. **Well-motivated problem.** The paper addresses the challenge of recovering system dynamics from stationary, intervention-only data, a realistic and important setting for many scientific domains (e.g., biology), where collecting time-series trajectories is often infeasible. 2. **Novel theoretical contribution.** The work provides, to the best of my knowledge, the first provable identifiability guarantees for SDEs observed only through stationary interventional distributions, covering both lin
1. **Theoretical presentation lacks clarity.** The main theorems are difficult to follow because they do not explicitly list all required assumptions. For instance, Theorem 4.4 depends on distributional/genericity assumptions (Assumption 4.2) and a known $D$ (Assumption 4.3), yet these are not stated in the theorem itself but scattered in the text. This weakens the precision and reproducibility of the claims. 2. **Restrictive linear setup.** The linear identifiability results rely on strong a
• Rigorous mathematical development with clear proofs. • Theoretical results are novel within the causal inference/SDE literature. • Experiments confirm the identifiability thresholds.
• The setting (identifiability from stationary SDEs) is quite narrow and primarily of mathematical interest. • The nonlinear result holds only under restrictive assumptions (contractive drift, small noise). • No real connection is made to learning algorithms or generative diffusion models, which would be essential for ICLR relevance. • The applications is minimal and does not add conceptual depth.
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Taxonomy
TopicsScientific Measurement and Uncertainty Evaluation