A Continuous-Time Ensemble Kalman-Bucy Smoother for Causal Inference and Model Discovery

Data assimilation (DA) integrates observational information with model predictions to improve state estimation in complex systems. While filtering provides the basis for online forecasts by using only past and present observations, it can exhibit delays and biases when the underlying dynamics evolve rapidly or undergo regime transitions. Smoothing, which additionally incorporates future observations, provides a natural pipeline for hindcasting and reanalysis that yields an uncertainty reduction beyond the filter. This paper introduces an ensemble Kalman-Bucy smoother (EnKBS) for continuous-time DA of nonlinear dynamical systems, where the smoother's conditional distributions are reconstructed using ensemble moments. The result is a derivative-free framework that does not require explicit computation of tangent-linear or adjoint models, which converges to the exact smoother solution at the infinite-ensemble limit for a wide class of complex systems. Incorporating standard regularization techniques for high-dimensional systems, such as covariance localization and inflation, the skill of the EnKBS is demonstrated in various important scientific problems. By integrating future observations, which reveal the underlying causal mechanisms for retrospective state updates, the EnKBS is used for Bayesian-based inference of causal relationships and their temporal influence range in a dyadic trigger-feedback model and the development of a causality-driven iterative learning algorithm that identifies the structure and recovers the hidden parameters of a nonlinear reduced-order model mimicking midlatitude atmospheric circulation. Notably, both tasks remain effective with an ensemble size of $O(10)$ under partial observations, suggesting that EnKBS can support the instantaneous discovery of high-dimensional complex systems over time.

Assimilative Causal Inference

Causal inference determines cause-and-effect relationships between variables and has broad applications across disciplines. Traditional time-series methods often reveal causal links only in a time-averaged sense, while ensemble-based information transfer approaches detect the time evolution of short-term causal relationships but are typically limited to low-dimensional systems. In this paper, a new causal inference framework, called assimilative causal inference (ACI), is developed. Fundamentally different from the state-of-the-art methods, ACI uses a dynamical system and a single realization of a subset of the state variables to identify instantaneous causal relationships and the dynamic evolution of the associated causal influence range (CIR). Instead of quantifying how causes influence effects as done traditionally, ACI solves an inverse problem via Bayesian data assimilation, thus tracing causes backward from observed effects with an implicit Bayesian hypothesis. Causality is determined by assessing whether incorporating the information of the effect variables reduces the uncertainty in recovering the potential cause variables. ACI has several desirable features. First, it captures the dynamic interplay of variables, where their roles as causes and effects can shift repeatedly over time. Second, a mathematically justified objective criterion determines the CIR without empirical thresholds. Third, ACI is scalable to high-dimensional problems by leveraging computationally efficient Bayesian data assimilation techniques. Finally, ACI applies to short time series and incomplete datasets. Notably, ACI does not require observations of candidate causes, which is a key advantage since potential drivers are often unknown or unmeasured. The effectiveness of ACI is demonstrated by complex dynamical systems showcasing intermittency and extreme events.