Identifiability Analysis of Linear ODE Systems with Hidden Confounders (2410.21917v2)

Abstract: The identifiability analysis of linear Ordinary Differential Equation (ODE) systems is a necessary prerequisite for making reliable causal inferences about these systems. While identifiability has been well studied in scenarios where the system is fully observable, the conditions for identifiability remain unexplored when latent variables interact with the system. This paper aims to address this gap by presenting a systematic analysis of identifiability in linear ODE systems incorporating hidden confounders. Specifically, we investigate two cases of such systems. In the first case, latent confounders exhibit no causal relationships, yet their evolution adheres to specific functional forms, such as polynomial functions of time $t$. Subsequently, we extend this analysis to encompass scenarios where hidden confounders exhibit causal dependencies, with the causal structure of latent variables described by a Directed Acyclic Graph (DAG). The second case represents a more intricate variation of the first case, prompting a more comprehensive identifiability analysis. Accordingly, we conduct detailed identifiability analyses of the second system under various observation conditions, including both continuous and discrete observations from single or multiple trajectories. To validate our theoretical results, we perform a series of simulations, which support and substantiate our findings.

• The paper presents a novel framework for determining identifiability in linear ODE systems with hidden confounders, extending standard analyses of fully observable cases.
• It transforms systems with latent variables into equivalent fully observable models through polynomial time transformations and DAG-based structures.
• Simulation results reveal that precise parameter estimates hinge on correctly modeling latent dependencies, thereby enhancing causal inference in dynamic systems.

Identifiability Analysis of Linear ODE Systems with Hidden Confounders

The presented paper, authored by researchers at the University of Melbourne and the University of California, San Diego, provides an in-depth exploration of the identifiability of linear Ordinary Differential Equation (ODE) systems that incorporate hidden confounders. The principal aim is to extend the current understanding of identifiability in these systems beyond the field of fully observable systems.

Identifiability is a crucial preliminary step in making robust causal inferences from ODEs, which are extensively used to model dynamic systems across various scientific fields such as physics, biology, and economics. While identifiability is extensively researched in fully observable systems, the presence of latent variables introduces significant complexity to this problem. This paper addresses this gap by presenting a framework to analyze the identifiability of linear ODE systems with hidden confounders.

The authors explore two primary scenarios within this framework. The first involves systems with independent latent confounders, where the evolution of these confounders follows predefined functional forms. They propose that identifiability can be determined when latent transformations like polynomial expressions of time are considered. In particular, they place emphasis on how identifiability is influenced by augmenting the observable system with additional parameters to account for the latent confounders.

The second, more complex scenario generalizes the first by introducing causally interdependent latent confounders, whose causal structures are captured by Directed Acyclic Graphs (DAGs). In these systems, the authors articulate a novel identifiability criterion leveraging the system's structure and the dynamics imposed by DAGs. They provide thorough analyses of systems under varied observation conditions—be it continuous or discrete, across single or multiple trajectories.

An interesting methodological contribution of this work is the transformation of systems with hidden confounders into equivalent fully observable systems. By doing so, they enable the application of existing identifiability results for standard homogeneous linear ODE systems. This approach allows for the derivation of identifiability conditions based on whether the augmented system, now devoid of latent variables, spans the full state space.

The authors support their theoretical findings with simulations, which substantiate their proposed identifiability conditions. The simulation results show that correct parameter estimates are often sensitive to the underlying structural assumptions being met. For instance, the presence of DAG in latent variables ensures distinguishability of system trajectories, emphasizing the importance of correctly modeling latent dependencies.

The implications of this work are vast, affecting both the theoretical understanding and practical applications of ODE systems in modeling causality. The theoretical contributions provide a foundation for designing estimators in latent ODE models, potentially influencing experimental designs in fields where controlled intervention on latent variables is possible. Practically, these results could be instrumental in improving the estimates of dynamic models in various applications where unobservable dynamics play a critical role.

This work elegantly bridges a significant gap in the identifiability of ODE systems with hidden confounders, offering a methodology that promises to enhance the reliability of causal inferences in dynamic systems. Future directions may extend this work to nonlinear systems or stochastic differential equations, further broadening the impact of these findings in the field of causal inference.