Assimilative causal inference

Domain: earth_science
Published: 2026-01-22
Editorial outcome: major_revision

Abstract
Causal inference is fundamental across scientific disciplines, yet existing methods struggle to capture instantaneous, time-evolving causal relationships in complex, high-dimensional systems. In this paper, assimilative causal inference (ACI) is developed, which is a methodological framework that leverages Bayesian data assimilation to trace causes backward from observed effects. ACI solves the inverse problem rather than quantifying forward influence. It uniquely identifies dynamic causal interactions without requiring observations of candidate causes, accommodates short datasets, and, in principle, can be implemented in high-dimensional settings by employing efficient data assimilation algorithms. Crucially, it provides online tracking of causal roles that may reverse intermittently and facilitates a mathematically rigorous criterion for the causal influence range, revealing how far effects propagate. The effectiveness of ACI is demonstrated by complex dynamical systems showcasing intermittency and extreme events. ACI opens valuable pathways for studying complex systems, where transient causal structures are critical.

Introduction
Causal inference determines cause-and-effect relationships between variables. It has found wide applications across different disciplines such as atmospheric and ocean science, economics, and neuroscience. In addition to discovering the interactions between variables, causal inference plays a significant role in model identification, policy evaluation, and decision-making.

Methods for causal inference can be classified into different categories depending on the available resources for use. On the one hand, a natural way to identify the overall causal structure is to exploit multivariate time series, where temporal dependencies are often utilized to infer causal relationships. Methods falling into this category include Granger causality, transfer entropy, mutual information, convergent cross mapping, causation entropy, and mixture causal discovery. On the other hand, models built upon physics can assist in understanding causal dependence between different variables. Information transfer based on the ensemble forecast of the underlying model has been exploited to indicate certain causal relationships over short terms. In addition, linear response theory, which infers causal links by analyzing system responses to small perturbations at the equilibrium, has been utilized to reveal the attributions of variations in climate systems. Other recently developed methods include using Koopman operators for causal discovery, causal graphs, dynamic neural relational inference, and approaches based on machine learning.

Nature can be regarded as a complex dynamical system where only a single random realization from this system is available. Due to the underlying strong nonlinearity and multiscale features, such a realization is usually intermittent and stochastic. Detecting instantaneous causal relationships as a function of time is then crucial for understanding the dynamic interplay among variables, where the roles of causes and effects can shift repeatedly over time and at irregular intervals. In environmental science, extreme events, such as the triggering mechanisms of hurricanes and the abrupt transitions in weather patterns, have significant scientific and social impact. Understanding the precursors of each extreme event and its subsequent implications requires online tracking of the causal relationship between them. Likewise, instantaneous causal analysis is of broad interest in neuroscience to discover temporary role reversals between brain regions during decision-making or cognitive tasks. However, most purely data-driven causal inference methods exploit data points over a long time series to reveal the average causal directions. In contrast, purely model-based methods, such as computing the information transfer in, utilize ensemble forecasts to infer the causal relationship when the system begins from a given initial condition and evolves toward statistical equilibrium. Yet, these methods are typically only able to provide time-dependent results for a very short time and become computationally challenging as the dimension of the system increases. Additionally, these model-based methods often struggle to optimally handle cases with a single observed realization.

Given the importance of discovering instantaneous causal relationships, this paper introduces a methodological framework called assimilative causal inference (ACI) to compute time-dependent causal interactions in complex systems. As in many practical applications, only a single realization of a subset of the state variables is available as observations. Meanwhile, an associated model, often turbulent and stochastic, is accessible as a supplement. Fundamentally different from traditional causal inference methods, which treat causality as a forward problem (analyzing how causes propagate into effects), ACI instead addresses it as an inverse problem. Using Bayesian data assimilation, ACI traces causes backward from observed effects by quantifying whether incorporating information about the effects reduces uncertainty in reconstructing the potential causes.

ACI has several desirable advantages. First, it captures the evolving interplay among state variables, as their causal and effect roles can be reversed across time. Thus, ACI provides direct insights beyond time-averaged causal links, making it particularly valuable for studying turbulent systems with intermittency and regime switching. Second, ACI identifies the causal inference range (CIR) of a relationship, revealing how far causal effects propagate at each time instant. The importance of the CIR in complex networks has been highlighted, ACI offers a unique and general method for computation in turbulent systems. Third, ACI efficiently handles high-dimensional problems by leveraging computational methods developed in Bayesian data assimilation. This is fundamentally different from information-transfer methods limited to low dimensions. In addition, ACI can accommodate short time series and incomplete datasets, which often appear in geophysics and climate science. Notably, ACI does not require observations of candidate causes, which is a key advantage since potential drivers are usually unknown or unmeasured. Instead, uncertainty reduction in potential causal variables can be determined solely from the observed effects and governing dynamical model via Bayesian data assimilation. As a final remark, while data assimilation has been used to assist in detecting the attribution of variations in climate systems, specific extreme events, and perturbations, rather than online causal inference or CIR estimation. ACI also fundamentally differs from these methods in its integration of data assimilation into causal inference.

The remainder of this paper is organized as follows. Numerical results from applying ACI on various nonlinear complex dynamical systems with intermittency and extreme events are presented in Section 2 (Results). Discussions and remarks on the developed methodology are contained in Section 3 (Discussion). The operational details of the ACI framework are included in Section 4 (Methods): Section 4.1 presents the mathematics of the ACI framework, followed by the discussion of CIR in Section 4.2. Section 4.3 extends the ACI framework to account for the presence of non-target variables. Mathematical derivations and additional test results are available in the Supplementary Information.

Results
We apply the ACI framework to identify causal relationships across the state space and evaluate the corresponding CIRs in complex nonlinear dynamical systems exhibiting intermittency and extreme events. The dynamical properties of these systems and the mechanisms that trigger crucial observed phenomena are well understood. This knowledge is exploited as guidance to validate the results when applying ACI. In addition, a perfect-model setup is used in the case studies carried out, meaning the system generates the synthetic observational data and the forecast model in ACI at the same. This facilitates the recovery of causal relationships in the absence of model error. Because the underlying system is chaotic and subject to stochastic forcing, a perfect model means that the model used in ACI shares the same governing equations and long-term statistical behavior as the system that generates the true signal. Additional results, including further numerical experiments, case studies, and one crucial application using sparse real-world observations, are provided in the Supplementary Information.

The operational outline of the ACI metric used to infer instantaneous causal relationships in the following experiments, including the details of how we objectively determine the associated CIRs and account for non-target variables, is described in Section 4 (Methods). The developed framework is also summarized via the schematic diagram provided in Fig. 3. Additional mathematical results and analyses, as well as computational implementation details, can be found in the Supplementary Information.

An nonlinear dyad model with intermittent extreme events
Let us start with a two-dimensional model, which nevertheless has strong nonlinear features with observed extreme events. The model reads as:

dx = (-d_x + \gamma y)x + f_x + \sigma_x \dot{W}_x, (1a)

dy = (-d_y - \gamma x^2)y + f_y + \sigma_y \dot{W}_y, (1b)

where \dot{W}_x and \dot{W}_y are independent (in time and mutually) standardized normal random variables. Within the proposed ACI framework, only x is assumed to be observed over time. This is a reduced-order conceptual model for atmospheric variability. It has been used to analyze the effects of various coarse-grained procedures on processes exhibiting intermittency, large-scale bifurcations, and microscale phase transitions. It is defined by an energy-conserving condition on its quadratic nonlinearities. The following parameter values are used for this model:

d_x = 0.5, d_y = 0.5, \gamma = 2, f_x = 1, f_y = 0.5, \sigma_x = 0.5, \sigma_y = 0.5, (2)

Figure 1 illustrates the ACI and CIRs from y to x. Panel (a) shows that extreme events in x occur intermittently. When the combined coefficient -d_x + \gamma y in (1a) becomes positive, the dynamics of x exhibit anti-damping, leading to amplitude growth and extreme events. Conversely, when the coefficient is negative, x behaves as a damped process, with fluctuations primarily driven by random noise. Panel (b) displays the ACI, revealing that phases with significant ACI values largely coincide with the onset and peak phases of extreme events in x. This aligns with intuition, as y acts as an anti-damping source and the primary driver of these events. Once x peaks and begins to decay, the ACI value drops sharply, reflecting the strong negative feedback from x to y, which then dampens x. The whisker plot in Panel (a) highlights the objective CIR, indicating a sustained influence from y to x during the triggering phase of extreme events but minimal influence during their demise. Notably, the long-range influence from the CIR reveals that extreme events develop gradually, with triggering conditions established well in advance. From a data assimilation perspective, when the observable signal x starts to strengthen, it enhances the ability of the smoother to estimate y as the signal in the future contains useful information beyond noise. As a result, strong ACI and longer CIR values are obtained. In contrast, once x becomes sufficiently strong, its high signal-to-noise ratio ensures the filter captures all relevant triggering dynamics, leaving little room for the smoother to improve estimates. Hence, the outset of the ACI metric’s decline and shorter CIRs. The Supplementary Information contain the filter and smoother distributions and the subjective CIRs with different threshold values \epsilon.

A stochastic model capturing the El Niño–Southern Oscillation (ENSO) diversity
El Niño–Southern Oscillation (ENSO) is a dominant climate phenomenon characterized by quasi-regular periodic behaviors in sea surface temperatures (SSTs) and in the atmospheric circulation across the equatorial Pacific Ocean. ENSO exhibits remarkable diversity in its spatial patterns, temporal evolution, and impacts, which can be broadly categorized into two main types: Eastern Pacific (EP) and Central Pacific (CP) El Niños, where an anomalous warming center occurs in the eastern and central Pacific, respectively. The opposite phases with anomalous cooling SSTs are called La Niña. Understanding the causes of different ENSO events is crucial for improving climate predictions and mitigating socio-economic consequences.

Although few models can accurately capture ENSO diversity, a recently developed stochastic conceptual model successfully reproduces its diverse behaviors and non-Gaussian statistics. This model has been highlighted in a recent review, making it a suitable testbed for studying the ENSO diversity. Mathematical details are provided in the Supplementary Information. The model consists of six state variables: ocean zonal current in the CP (u), western Pacific (WP) thermocline depth (h_W), CP SST (T_C), EP SST (T_E), atmospheric winds (\tau, intraseasonal), and background Walker circulation (I, decadal). The first four variables (u, h_W, T_C, and T_E) operate on interannual time scales. As an nonlinear system with state-dependent noise, the model generates extreme events and intermittency. The two SST variables allow reconstruction of spatiotemporal SST patterns across the equatorial Pacific, providing an intuitive way to identify different ENSO event types.

The variables T_C, h_W, and \tau directly influence T_E. Figure 2 displays their conditional ACI and CIR values to T_E (non-target variables have been resolved via the framework proposed in Section 4.3). For EP El Niño events (positive T_E anomalies, shown in red), T_C exhibits the strongest ACI value, slightly preceding the T_E peak. This timing aligns with physical understanding, as SSTs in these regions are strongly coupled: during El Niño, warm water propagates from CP to EP, producing the observed ACI lead. The \tau to T_E ACI value appears noisier due to \tau’s intraseasonal variability, yet its evolution confirms \tau’s robust impact on T_E. Winds drive warm water propagation and exert near-instantaneous SST effects. In contrast, while h_W significantly contributes to T_E, its ACI amplitude is weaker than T_C or \tau. The discharge–recharge theory posits h_W – T_E oscillator dynamics, but in models with refined CP variables, h_W’s influence on T_E becomes indirect: h_W first affects T_C, which then propagates signals to T_E (WP \rightarrow CP \rightarrow EP). Consistent with this paradigm, h_W’s ACI value to T_E peaks months before EP El Niño maxima. CIRs further corroborate these physical mechanisms: T_C shows the longest influence, h_W’s more indirect role yields intermediate CIRs, and \tau’s intraseasonal nature produces the shortest impacts. The Supplementary Information provides additional ACI and CIR analyses for all variables across different ENSO event types. In addition, in the Supplementary Information, a case study using real-world ENSO observations is examined, which does not assume a perfect model setup. The conclusions derived about the causal relationships using real-world observational data remain qualitatively similar to those with model-generated data. The results provide preliminary evidence that the ACI framework remains reasonably robust under realistic conditions, including observational noise, moderate model error, and numerical discretization effects.

Methods
In this section, we provide a high-level overview of the methodology underlying the ACI framework, including its formulation of causal relationship assessment through a Bayesian inverse problem and the evaluation of the associated CIRs via objective means. We also outline the strategy used to infer causal links while accounting for interference from non-target or ancillary variables. Rigorous mathematical justifications are provided in the Supplementary Information.

The assimilative causal inference (ACI) framework
Setup and notation. Consider a time interval [0, T] and a specific time instant t, where 0 \le t \le T. Let {x(t)}_{0 \le t \le T} and {y(t)}_{0 \le t \le T} be two multivariate stochastic processes defined over this interval. Denote by x(t) and y(t) the corresponding multivariate random variables at time t. Their evolution in time is defined by the following stochastic dynamical system:

dx = f_x(t, x, y) + \Sigma_x(t, x) \dot{W}_1(t) + \Sigma_x(t, x) \dot{W}_2(t),

dy = f_y(t, x, y) + \Sigma_y(t, x, y) \dot{W}_1(t) + \Sigma_y(t, x, y) \dot{W}_2(t),

where \dot{W}_1 and \dot{W}_2 are independent (in time and mutually) mean-zero Gaussian random vectors with uncorrelated components that possess unit variance. Mathematical details regarding the structure of (3) are given in the Supplementary Information. For notational simplicity, the dependence on t is sometimes omitted when referring to the state variables and no distinction between random variables and their realizations is made. Let x(s \le t) represent one realization of the stochastic process {x(t)}_{0 \le t \le T} as a time series over the interval [0, t]. Similarly, x(s \le T) denotes a realization of {x(t)}_{0 \le t \le T} as a time series over the entire interval [0, T]. For simplicity, the time series are assumed to be continuously observed. An analogous framework can be developed for discrete-in-time observations. Throughout this paper, the time series is assumed to be free of observational noise.

According to Granger’s predictive causality or its nonlinear extension, transfer entropy, y is the cause of x at time t if the knowledge of y(s \le t) improves the prediction or reduces the uncertainty of x(t). Using information theory, this can be expressed as:

S(p(x(t)|y(s \le t))) < S(p(x(t))),

where S(\cdot) is Shannon’s entropy and the history of the target variable in the conditioning, i.e., x(s < t), is omitted in (4) for notation simplicity.

Causal discovery from a Bayesian inverse problem viewpoint. The above argument can be interpreted in a reverse way, from the perspective of statistical inference through an inverse problem in uncertainty quantification. If x is the subsequent effect of y(t), then knowing future information about x reduces the uncertainty in inferring the current state of y(t):

S(p(y(t)|x(s > t))) < S(p(y(t))).

Further, including the past and current information of x on both sides yields the filter and smoother estimates. The Supplementary Information contains more technical details.

S(p(y(t)|x(s \le T))) < S(p(y(t)|x(s \le t))),

which does not change the causal relationship in (5). A notable feature of (6) is that it establishes a connection between causal discovery and Bayesian data assimilation. Suppose the underlying turbulent and potentially stochastic model governing x and y is known and given by (3). Running the model forward provides a statistical estimation of the state of y(t) (known as ‘forecast’). However, this state estimation differs when also incorporating the knowledge of the time series x(s \le t) or x(s \le T) (known as ‘analysis’), as these specific observations provide additional information to reduce the uncertainty when inferring the state of y(t). This two-step (forecast-analysis) pipeline is known as Bayesian data assimilation, where the distribution derived solely from the model statistics is referred to as the prior, while the observed time series is used to compute the likelihood, accounting for the uncertainty in the observational process. The two conditional distributions in (6) result from combining the model-based information with the observation-induced likelihood and are referred to as the posteriors. Specifically, the distribution on the left-hand side of (6) corresponds to smoothing, while the one on the right-hand side corresponds to filtering.

ACI. Denote by p_t^s and p_t^f the smoother and filter posterior distributions at time t (i.e., the two distributions on the left- and right-hand sides of (6)), respectively. If the uncertainty reduction in p_t^s is more significant than that in p_t^f, then it is due to the incorporation of the future information of x, which indicates toward the contribution from y(t) to the future states of x.

Although the entropy difference has been widely used in many other causal inference methods, the relative entropy is employed hereafter to quantify the uncertainty reduction in p_t^s related to p_t^f. In addition to the covariance (and other higher-order moments), which measure the variability of the random variable, the relative entropy also accounts for the difference in the mean state of the two distributions. This difference is crucial for reflecting the additional information gained by incorporating future observations. Furthermore, it is coordinate-free, i.e., it is invariant under general nonlinear changes of the state variables. Notably, relative entropy has been widely used to assess uncertainty reduction in the context of data assimilation.

As such, if the relative entropy between the smoother and filter distributions is nonzero, i.e.,

P(p_t^s, p_t^f) = \int p_t^s \ln(p_t^s / p_t^f) > 0,

then y is identified as the cause of x at time t. By computing (7) at different t, a time-dependent causal link is established.

Under (7), ACI provides a unique formulation for identifying causal relationships. In traditional approaches, causal links are inferred by running (3) forward to quantify the informational response on the target variables’ entropy when conditioned on the history of the candidate causes. ACI instead solves an inverse problem, thus tracing causes back from the data of the observed subsequent effects. By interpolating causality from effects to causes instead of extrapolating the latter forward-in-time to identify the former, ACI offers an inherently more robust framework compared to classical predictive causality. Interpolation is typically more stable because it operates within the range of observed information, whereas extrapolation requires projecting beyond available data and is therefore far more sensitive to model error and uncertainty.

Dynamic discovery of the causal influence range (CIR)
While the uncertainty reduction in the smoother solution related to the filter solution in (7) indicates an instantaneous causal link from y to x at time t, it does not reveal the temporal extent of this causal influence, namely, how many future units of x are affected by y(t). Since the memory of turbulent systems decays over time, y(t) is expected to effectively influence the future states x only within a limited time period, where this causal influence range can vary significantly as a function of t. This raises two key questions: How can we determine the CIR and, more importantly, how can we do so objectively, i.e., without relying on empirical cutoff thresholds? This section outlines the core ideas for addressing these questions. Rigorous mathematical analysis and detailed algorithms are provided in the Supplementary Information.

On-line smoothing for dynamic influence tracking. Recall from (7) that the relative entropy P(p_t^s, p_t^f) between the filter and smoother estimates quantifies the causal influence from y(t) to x. Here, p_t^s = p(y(t)|x(s \le T)) incorporates the complete future information of x. Due to the finite effective influence length, the complete smoother solution p_t^s can be achieved by utilizing future information up to t \le T' \le T, for a short time period after t. The divergence between these distributions vanishes as T' \rightarrow T. We therefore define the causal influence radius or CIR through a relative entropy criterion: For a predetermined threshold \epsilon \ge 0, the maximum lag T'_\epsilon \in [t, T] satisfying

P(p(y(t)|x(s \le T)), p(y(t)|x(s \le T'_\epsilon))) > \epsilon,

yields the innate CIR measure \tau_\epsilon(t) = T'_\epsilon - t. This adaptive time variable \tau_\epsilon(t) defines the effective causal period [t, t + \tau_\epsilon(t)] during which y(t) exerts measurable influence on x. A recently developed online smoother algorithm is employed to compute the CIR. This online smoother computes the uncertainty reduction in the estimated state of y(t) when the future information of x is sequentially added. See the Supplementary Information for more technical details.

While the CIR computed via (8) is intuitive, it remains inherently subjective due to its dependence on the threshold parameter \epsilon. In the following, an objective CIR formulation is developed, eliminating the need for a prescribed threshold. This threshold-independent measure is subsequently integrated into the ACI framework.

Objective CIR. For simplicity, assume that the relative entropy between the aforementioned complete and lagged smoother distributions decreases monotonically as more future information of x is incorporated. This commonly holds in practice, though the theory of the objective CIR (developed rigorously in the Supplementary Information) doesn’t require this condition. At a given time instant t, let M = P(p_t^s, p_t^f) denote the maximum value of the relative entropy between the complete and lagged smoother distributions over t \le T' \le T, which is the difference between the smoother estimate and the filter estimate (where the latter can be viewed as a degenerate smoother using no future information). Given a threshold \epsilon \in [0, M], (8) yields a subjective CIR as \tau_\epsilon(t) = T'_\epsilon - t. The objective CIR is then naturally defined by

\tau(t) = \frac{1}{M} \int_0^M \tau_\epsilon(t) d\epsilon.

Dividing by M guarantees the unit of the objective CIR is ‘time’. See Panel (d) of Figure 3 for a schematic illustration of the objective CIR and its relationship with the subjective ones. Note that numerically evaluating the integral in (9) requires repeated smoother computations, leading to a computational complexity that scales quadratically with the number of discretization points. An efficient computational method for (9) is included in the Supplementary Information.

The subjective and objective CIRs can be regarded as analogs of the autocorrelation function and decorrelation time, respectively. The autocorrelation function measures the memory of a turbulent system based on the duration of its values that remain above a predetermined threshold. However, determining the memory using the autocorrelation function is subjective and can vary significantly depending on the threshold value. In contrast, the decorrelation time, which integrates the autocorrelation function, provides an objective way to define the memory length and is free from any predetermined cutoff threshold.

Conditional ACI in the presence of additional non-target variables
In the more general context of a complex system, state variables can be grouped into three disjoint subsets: x_A, x_B, and y, with x = (x_A, x_B)^T under the previous notation. The goal is to determine whether y is a cause of x_A in the presence of the remaining variables x_B, which are called the non-target variables, including the cases of confounders or mediators. Since x_B interacts with both x_A and y in the dynamics, appropriately accounting for the role of x_B while inferring the causal link from y to x_A becomes essential. Hereafter, let us assume that a realization for each of x_A and x_B, namely x_A(s \le T) and x_B(s \le T), is available. Therefore, the candidate cause y is the only variable that does not require to have observations in this framework. Following the argument in Section 4.1, the causal relationship from y(t) to (x_A, x_B) is identified based on the uncertainty reduction in the smoother distribution p_t^s related to the filter one p_t^f, where

p_t^f = p(y(t)|x_A(s \le t), x_B(s \le t)),

p_t^s = p(y(t)|x_A(s \le T), x_B(s \le T)).

The main focus here is to appropriately determine and resolve the presence of x_B from p_t^f and p_t^s as to infer a conditional causal relationship from y(t) to x_A beyond the contributions of x_B.

Recall from Section 4.1 that ACI employs Bayesian data assimilation to estimate the state of y(t). ACI exploits observations of x to reduce the uncertainty in y(t), where the influence of x on the estimation of y(t) depends on its own uncertainty: smaller observational uncertainty leads to a stronger impact. When extending this framework to x = (x_A, x_B)^T for conditional ACI, the observed time series of x_B is still used in the forecast step to compute the uncertainties in x_A and y. However, during the analysis step, the uncertainty in x_B’s dynamics is deliberately excluded by assigning infinite uncertainty to its marginal likelihood before posterior computation. This can be rigorously formulated through a limit operation on the analysis component of the Bayesian data assimilation update equations, which formally leads to the following filter- and smoother-based distributions:

p_t^f|x_B := \lim_{Var(x_B(t)) \rightarrow +\infty} p_t^f, p_t^s|x_B := \lim_{Var(x_B(t)) \rightarrow +\infty} p_t^s.

Similar to (7), if the relative entropy of p_t^s|x_B from p_t^f|x_B is positive, P(p_t^s|x_B, p_t^f|x_B) > 0, then y is identified as the cause of x_A conditional to x_B at time t. This means future data of x_A incur additional information gain about y(t) related to its history, beyond the observational contributions of x_B. By computing this metric at each t, a time-dependent conditional causal relationship is established.

Inflating the uncertainty in x_B is straightforward using Bayesian inference for state estimation. This approach ensures that x_B does not affect the uncertainty reduction in the state estimation of y(t) when treated as an observational process. Consequently, any uncertainty reduction in y(t) results solely from x_A under this proposed pipeline. See Panel (c) of Figure 3 for an illustration. For dynamical systems with explicit governing equations (see (3)), a shortcut to exclude the influence from the observational uncertainty of x_B is to treat x_B as a prescribed forcing term in the reduced system defined by (x_A, y) during Bayesian inference, with values defined by its observed time series. Nevertheless, this framework does not necessarily require modifications to the underlying dynamical system, making it compatible with any given model, potentially including operational models in atmospheric and ocean science, as the model structure and integrity remains intact.

It is important to emphasize that while treating x_B as a conditioning variable may appear analogous to the role of non-target variables in transfer entropy and other traditional causal inference methods, the way of handling the uncertainty in ACI is fundamentally different. Conventional methods treat x_B as a fixed component. In contrast, ACI explicitly accounts for x_B’s dynamical influence while systematically prohibiting its uncertainty from affecting state estimation. This unique treatment allows the inferred causal relationships to reflect true dynamical interactions by isolating the contributions on the uncertainty reduction from the non-target variables. As a final remark, if Bayesian data assimilation is used to compute p(y(t), x_A(t)|x_B(s \le T)) and then marginalize over x_A(t) to obtain p(y(t)|x_B(s \le T)), this manipulation may lead to incorrect conclusions. This is because the correlation between y(t) and x_B(t) unavoidably alters the uncertainty in y(t) during Bayesian data assimilation, potentially introducing a spurious causal relationship between y and x_A. This is true even if we condition over x_B(t) to obtain p(y(t)|x_B(s \le T), x_A(t)). For instance, for the causal chain y \rightarrow x_B \rightarrow x_A, such a method might falsely imply a direct causal link from y(t) to x_A.

The details of the conditional ACI implementation, along with the verification of (conditional) nil causality principles under ACI, are provided in the Supplementary Information.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Key results\nThe authors introduce Assimilative Causal Inference (ACI), a principled framework for uncovering causal relationships in\ndynamical systems using Bayesian data assimilation. A key strength of ACI is its ability to infer causal structure from a single\nrealization, even when the underlying causal relationships evolve over time. Specifically, ACI adopts a Bayesian Inverse\nProblem perspective, asking whether knowledge of future observations helps reduce uncertainty about potential past\ncauses. The authors also develop efficient algorithms to compute the causal influence range (CIR)—the time window over\nwhich a cause significantly impacts an effect. The method is demonstrated on simulated dynamical systems, highlighting its\nability to recover transient and time-varying causal patterns.\nValidity\nThe authors test the ACI framework on synthetic datasets generated by well-known dynamical systems, under the\nconditional gaussian nonlinear systems (CGNS) assumptions on the data generation process. While the analysis of the\nresults is sound and highlights the key features of the method well, there is no explicit accuracy measure of the inferred\ncausal influences or CIRs. Comparing ACI’s outputs against a ground-truth dataset with explicitly known causal influences\nand CIRs would help validate the method’s accuracy and address concerns about the reliability of its inferred causal\nrelationships and CIR estimates.\nThe manuscript’s claim of scalability to high-dimensional systems is not supported by theoretical analysis or empirical\nevidence. To substantiate this claim, the authors should either provide a discussion of the method’s computational\ncomplexity as a function of the number of variables, or demonstrate its practical performance on higher-dimensional\ndatasets. Currently, all experiments are limited to systems with at most six state variables, which does not convincingly\ndemonstrate scalability.\nSignificance\nThis work has potentially high significance, as it addresses several major limitations of existing temporal causal discovery\nmethods. In principle, ACI can accommodate instantaneous causal effects, latent confounders, and time-varying causal\nstructures, all from a single realization. However, these capabilities remain largely theoretical, as they are not fully explored\nor validated through empirical analysis. Additional experiments are needed to validate the robustness of the proposed\nmethod against these factors. Additionally, the manuscript lacks comparisons with established baseline methods, such as\ncausation entropy, transfer entropy, or convergent cross mapping (CCM), which limits the ability to assess ACI’s relative\nstrengths and practical utility.\nData and methodology\nAll experiments are conducted on low-dimensional, synthetic datasets generated by the authors, including benchmark\nsystems such as the nonlinear Dyad model and a simplified ENSO model. While these examples effectively demonstrate the\nmethod’s core ideas, they offer limited insight into ACI’s performance on real-world data, which typically involve higher\ndimensionality, observation noise, and potential violations of the CGNS assumptions. As no real-world datasets are\nanalyzed, and important factors like sensitivity to noise, model mismatch, or partial observability are not systematically\nexamined, it is difficult to assess the method’s robustness and generalizability. Including an experiment on a real-world\ndataset would substantially strengthen the methodological contribution.\nAnalytical approach\nThe analytical framework proposed in the paper is creative and well thought out. The derivation of estimators under the\nConditional Gaussian Nonlinear System (CGNS) assumption is detailed and appears sound, though I did not verify all low-\nlevel derivations. The use of infinite uncertainty for the non-target variables (i.e.., x_B) is a subtle but intuitive modeling\nchoice, and the justification provided aligns well with Bayesian data assimilation principles.\nHowever, the framework’s heavy reliance on the CGNS assumption needs to be examined in more detail. Since the\ntractability of ACI hinges on this assumption, it would be helpful to understand how critical it truly is. How does performance\ndegrade when the system violates this condition? An empirical test of the method under such violations would strengthen\nthe argument for its broader applicability.\nAdditionally, while ACI offers a fresh perspective by tracing causes backward from observed effects, it appears conceptually\nsimilar to transfer entropy and causation entropy, both of which also quantify the information flow between time-lagged\nvariables. A discussion contrasting ACI with these methods—particularly in terms of expressivity, computational complexity,\nand assumptions—would help position the contribution more clearly within the existing literature. Without such a\ncomparison, it remains unclear when and why ACI would be preferred over these well-established alternatives.\nClarity and context\nWhile the overall presentation is clear, certain technical aspects would benefit from additional context and clarification. For\nexample, the discussion in Section 4 regarding the use of infinite uncertainty for x_B is difficult to follow in isolation. Its\nrationale becomes clear only when cross-referenced with Supporting Information Sections 1.4.2 and 1.4.3.\nAdditionally, the computational procedure underlying ACI is fairly complex. Including a pseudocode algorithm block that\noutlines the major steps would improve accessibility for a broader audience and facilitate reproducibility.\nSuggested improvements\nValidation and Ground Truth Testing: Include experiments with synthetic datasets where the true causal influences and\nCIRs are explicitly known.\nScalability Assessment: Provide theoretical analysis of computational complexity as a function of system dimensionality,\nand demonstrate ACI's performance on higher-dimensional datasets.\nRobustness Analysis: Conduct systematic experiments examining ACI's sensitivity to key factors including observation\nnoise, violations of CGNS assumptions, and partial observability.\nBaseline Comparisons: Include comparative analysis with established methods such as causation entropy, transfer entropy,\nand convergent cross mapping (CCM).\nReal-world Application: Add at least one experiment on real-world data to demonstrate practical applicability beyond\nsynthetic examples.\nEnhanced Clarity: Include a pseudocode algorithm block outlining ACI's major computational steps to improve accessibility\nand reproducibility.\nReferences\nThe manuscript appropriately references most relevant works in the field. To strengthen the literature review, consider\nadding a citation to Graber and Schwing (2020) [1], which specifically addresses dynamic causal links using Granger\ncausality and provides methods for causal discovery from single realizations.\n[1] Graber, Colin, and Alexander Schwing. \"Dynamic neural relational inference for forecasting trajectories.\" Proceedings of\nthe IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops. 2020.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "Dear editor and authors,\nThis is my review of the manuscript entitled \"Assimilative Causal Inference\".\nThe manuscript introduces, formally defines and thoroughly tests a new method for inferring causality: Assimilative Causal Influence (ACI). ACI promises to detect causal influence under a variety of conditions, including high-dimensional systems, even in cases where the strength and direction of causality changes over time. These conditions are often requirements in fields such as climatology, ecology and neuroscience and presumably also highly relevant in other areas such as physiology and economics. In addition to ACI the paper introduces Causal Influence Range (CIR), which is a measure for the time-scale at which the the causal influence is effective. The authors circumvent an inherent ambiguity (subjectivity) in this measure by developing an objective version which is a normalized integral over all values of relative entropy.\nACI is based on inverse inference, i.e., knowing future values of the effect (at time later than t) reduces the uncertainty in inferring the value of the cause at time t. This is formalised using relative entropy (KL divergence) in a Bayesian data assimilation framework. The method is clearly explained in the main paper, with a more detailed explanation in the SI.\nFigure 2.1 is also very helpful for understanding ACI and objective CIR.\nMy overall assessment is that this is a very well-written paper and that the developed method is novel and will be highly interesting and relevant to many researchers. The authors present their method and results in a succinct and thorough way, and my recommendation is to publish the paper more or less as is, except for one suggestion and some minor issues (see below).\nMajor comment:\nI really only have one major suggestion, which is that I think the authors should make the code used for analysis and figures available up-front, instead of requiring interested readers to contact the corresponding author. I think this will be in the interest of the authors as well, since it will make the method more accessible. A second reason is that I have had several unfortunate experiences of writing authors to request code and/or data from papers and either have not received an answer or been rejected on various grounds. While I have no reason to think that the authors of the current manuscript would not be open to sharing, sometime people change jobs, e-mail addresses or even lose or misplace code and data. I sincerely hope the authors will follow this suggestion. This is a beautiful and well organized paper, and I think having the code would increase its usefulness to interested readers a lot.\nMinor comments:\n– In the abstract the authors write that ACI \"provides online tracking...\" I was initially confused about the use of the word online and prefer the formulation they use on page 1, line 39: \"real-time tracking of the causal relationship\" which, if I understand correctly, means the same thing.\n– On p. 5, l. 173 the wrong symbol is used. It should be $\\varepsilon$, not $\\epsilon$ for the threshold parameter.\n– On p. 7, l. 274 the same mistake recurs.\n– In section 2.3 the variable x is split into two subsets x_A and x_B, but this is not mentioned explicity (except in the SI). It would be helpful to state briefly in the main text as well, since x is used later on in the section to refer to the union of x_A and x_B (e.g. on p. 6, l. 234-235).\n– In section 3.1, Eq. (3.1a) and (3.1b): W_x and W_y (and their derivatives) are not defined or explained. Again the SI will help the reader, but a short comment here would help.\n– Figure 3.1: If space permits, it would be interesting to see a third panel with a plot of objective CIR as a function of t. I find it a bit difficult to decode the whisker plot in panel a.\nRegards,\nDan Mønster", "reviewer_4": "Review: Assimilative Causal Inference, by Andreou, Chen and Bollt\nThe authors introduce a framework for Assimilative Causal Inference (ACI), which harnesses Bayesian data assimilation to trace the candidate causes of observations. It is demonstrated to be effective for high-dimensional dynamical systems and short datasets, providing real-time tracking of causal roles and establishing a mathematically rigorous criterion for the causal influence range.\nThe authors describe the general significance of ACI across various scientific disciplines and some of the limitations of existing methods in capturing instantaneous, time-evolving causal relationships in complex systems. The approach is both powerful in scope of application and rigorous in a self-consistent sense and thus should be of interest to a broad scope of researchers interested in complex dynamical systems. In my opinion the paper should be published following a more sober assessment of its limitations and comparison/contrast to other approaches, such as those described in the second paragraph of the introduction. For example, I find it intellectually and emotionally distasteful for an author to claim their approach is “paradigm-shifting”, as is done in the second sentence of the abstract, upon its introduction? Paradigm shifts require a broad adoption and consensus in the community and years of testing and refinement. This concept was introduced by Thomas Kuhn in his “The Structure of Scientific Revolutions”, Univ Chicago Press, 1962. Paradigm shifts in science are not defined by fiat by an author and they take time to evolve. How do the authors know that next week, month or year, etc. another method will not appear that is far superior and supersede all others? Indeed, let the community and time decide what is paradigm shifting.\nThe paper highlights the potential of ACI to open new avenues for studying complex systems with transient causal structures and suggests further areas for research and application of ACI. It is one important method of identifying and tracing back causes from observed effects through Bayesian data assimilation and can identify interactions without direct candidate cause observations. It can provide real-time tracking and mathematically validated causal influence range and was shown to be effective in systems showcasing intermittency and extreme events.\nAs is noted, there are both general limitations and application-specific limitations, that make the paradigm-shifting claims specious. These include:\n(1) The fact that the ACI framework may be affected by the correlation between variables during Bayesian data assimilation, which can potentially introduce a spurious causal relationship, even if conditioned over intermediary variables. This can create erroneous interpretations of data.\n(2) The real-time tracking capability of the ACI framework necessitates continuous observation of a dataset. This requirement can be challenging in systems where data is not continuously available or is incomplete, which is often the case.\n(3) Although ACI claims to scale efficiently to high dimensions, the complexity of the framework still poses significant computational challenges, particularly in very large-scale systems.\n(4) The framework assumes the underlying models are correctly specified without observational noise, a condition that is rarely true in real-world applications where measurements are often noisy in many distinct, but difficult to quantify, ways.\n(5) While the authors noted the applicability in short datasets, the accuracy and reliability of causal inferences drawn from very limited data remain a concern.\n(6) The application-specific limitations include (a) extreme event analysis, where accurate parameter selection and model calibration are essential; (b) historical data limitations, such as shifts not represented in a dataset and (c) interference handling, which can compromise the reliability of the inferred causal relationships.\nWhile these limitations are noted, and notable, the reader will strongly benefit from knowing how these few, of potentially many more, are treated in other approaches in the field, many of which are mentioned in the introduction and are known to the authors in more detail."}, "author_response": "Response to Reviewers\nRef.: Response to Referees\nManuscript Number: NCOMMS-25-48940\nManuscript Title: Assimilative Causal Inference\nAuthors: Marios Andreou; Nan Chen; Erik Bollt\nCorresponding Author: Nan Chen\nDear Reviewers:\nWe deeply appreciate the reviewers for taking the time to thoroughly review our work and provide such\nthoughtful and insightful feedback. The reviewers’ detailed and discerning eye helped us clarify several\nvaluable points and implement additions and revisions that will further aid readers in understanding and\nappreciatingourframework. Wesincerelythankthereviewersfortheirtime,rigor,insights,andendorsement\nthrough the peer review process.\nTherevisedmanuscriptandSupplementaryInformation(SI)areattachedforyourconsideration. Allchanges\nare also clearly marked in the attached diff files, both for the main text and SI, which we upload alongside\nour revised manuscript and response letter. These files clearly indicate the changes made to our initial\nsubmission. Specifically, blue text indicates additions made in the revision, while strikethrough red text\nindicates removals.\nIn the following pages, we provide a detailed, point-by-point response to the reviewers’ comments, outlining\nhow each has been thoughtfully considered in the revised manuscript. At the end, we also list some addi-\ntional minor changes that have been made to the initial manuscript beyond the reviewers’ recommendations\nand suggestions. These include fixing minor typos, adjusting figures for clarity and readability, and making\nadditional clarifications to the exposition. These revisions do not alter the theoretical results or the deduc-\ntions made from the numerical case studies carried out in the initial submission. We hope that with these\nadditional changes and corrections, our manuscript is much more readable and accessible.\nWhat follows is a detailed account of how each of the reviewers’ points has been addressed in the revised\nmanuscript and Supplementary Information (SI).\nPoint-by-Point Response to Reviewers’ Comments:\nReviewer #1\nRemarks to the Authors:\nKey results\nThe authors introduce Assimilative Causal Inference (ACI), a principled framework for uncovering causal\nrelationships in dynamical systems using Bayesian data assimilation. A key strength of ACI is its ability\nto infer causal structure from a single realization, even when the underlying causal relationships evolve\nover time. Specifically, ACI adopts a Bayesian Inverse Problem perspective, asking whether knowledge of\nfuture observations helps reduce uncertainty about potential past causes. The authors also develop efficient\nalgorithms to compute the causal influence range (CIR)—the time window over which a cause significantly\nimpacts an effect. The method is demonstrated on simulated dynamical systems, highlighting its ability to\nrecover transient and time-varying causal patterns.\n▶ Authors’ Response: We sincerely thank the reviewer for taking the time to provide a very thorough and\nconstructive review of our paper. In what follows, we carefully outline how each concern raised has been ad-\ndressed in the revised manuscript. We believe that the changes made in line with the reviewer’s suggestions\nhave substantially improved the clarity, rigor, and overall quality of the paper. For that, we again thank the\nreviewer.\nValidity\nThe authors test the ACI framework on synthetic datasets generated by well-known dynamical systems,\nunder the conditional Gaussian nonlinear systems (CGNS) assumptions on the data generation process.\n▶ Authors’ Response: Thank you for pointing this out. We would like to offer an important clarification:\nalthough the numerical illustrations of ACI in this work use dynamical systems with the CGNS structure,\nthe ACI framework itself is developed for general nonlinear dynamical systems (see also Section 1.2.1 in the\nSI). The CGNS models were chosen as test cases because their closed-form analytic expressions in Bayesian\ndata assimilation eliminate the sampling errors that typically arise from ensemble methods. This allows\nus to focus on studying the ACI framework itself without the confounding influence of sampling or other\nnumerical errors. The general formulation of dynamical systems under the ACI framework is as follows:\ndx(t)=fx(t,x,y)dt+Σx(t,x)dW (t)+Σx(t,x)dW (t),\n1 1 2 2 (1)\ndy(t)=fy(t,x,y)dt+Σy(t,x,y)dW (t)+Σy(t,x,y)dW (t).\n1 1 2 2\nIn (1), we do not assume that fx or fy are linear in the unobserved variable y, or that the noise feedback\nmatrices Σy and Σy do not depend on y, which are the assumptions for a CGNS. The only assumptions\n1 2\nthat accompany (1) are the (I)–(V) regularity conditions noted in Section 1.2.1 of the SI. We understand\nthat this modeling assumption is not explicitly communicated to the reader in the main text and appears\nonly in the SI, which is a concern that Reviewer #3 has also raised. Therefore, we have added the following\nparagraph to the beginning of Section 2.1.1 in the main text to explicitly address that the ACI framework\ndeveloped here applies to general stochastic systems, not limited to CGNSs:\n“Consider a time interval [0,T] and a specific time instant t, where 0 ≤ t ≤ T. Let {x(t)} and\n0≤t≤T\n{y(t)} be two multivariate stochastic processes defined over this interval. Denote by x(t) and y(t) the\n0≤t≤T\ncorresponding multivariate random variables at time t. Their evolution in time is defined by the following\nstochastic dynamical system:\ndx\n=fx(t,x,y)+Σx(t,x)W˙ +Σx(t,x)W˙ ,\ndt 1 1 2 2\n(2.1)\ndy\n=fy(t,x,y)+Σy(t,x,y)W˙ +Σy(t,x,y)W˙ ,\ndt 1 1 2 2\nwhere W˙ and W˙ are independent mean-zero Gaussian random vectors with uncorrelated components that\n1 2\npossess unit variance. Mathematical details regarding the structure of (2.1) are given in the Supplementary\nInformation.’’\nWhile the analysis of the results is sound and highlights the key features of the method well, there is no\nexplicit accuracy measure of the inferred causal influences or CIRs. Comparing ACI’s outputs against a\nground-truth dataset with explicitly known causal influences and CIRs would help validate the method’s\naccuracy and address concerns about the reliability of its inferred causal relationships and CIR estimates.\n▶ Authors’ Response: Thank you very much for raising these important concerns. With respect to explicit\nCIRaccuracybenchmarks,tothebestofourknowledge,ACIisthefirstgeneralframeworkdesignedtoquan-\ntify the temporal reach of instantaneous causal impact in the predictive (future) direction. Consequently,\nit differs fundamentally from traditional time-averaged causal inference methods, in which the availability\nof large datasets enables the definition of a statistical ground truth and the implementation of standard\nvalidation procedures. In contrast, the goal of the ACI framework is to detect instantaneous causal relation-\nships. In this setting, the true signal consists of only a single data point at each time instant, which makes\nconventional statistical validation (such as direct comparison with ground-truth or theoretical CIR values)\nimpractical, even for relatively simple models. For this reason, such traditional validation is not included in\nthe present work.\nNevertheless, weintroducedthedyadmodelwithextremeeventsasourfirsttestexample. Sincethemecha-\nnismtriggeringextremeeventsisapparentfromthedynamicalstructure,weusedthisunderlyingdynamical\npropertyasaqualitativetruthtovalidateourACImethod. Similarly,intheSI,thewell-understoodfeedback\nloop mechanism in the predator–prey model is used to validate the inference results from the ACI method.\nWealsowishtopointoutthatseveralofthetestsinthisworkareconductedunderperfect-modelscenarios,\nmeaningthereisnomodelerrorwhenvalidatingtheACImethod. Inaddition,closed-formanalyticformulae\nfrom the CGNSs studied also eliminate sampling errors present in ensemble methods. These setups provide\nanidealizedenvironmentforvalidatingthemethod’saccuracy. Wehaveaddedthefollowingatthebeginning\nof the results section to clarify that we used these well-understood models to generate synthetic data for\nvalidating ACI and to point out the difference between the standard validation approaches:\n“The dynamical properties of these systems and the mechanisms that trigger crucial observed phenomena\nare well understood. This knowledge is exploited as guidance to validate the results when applying ACI. In\naddition, a perfect-model setup is used in the case studies carried out, meaning the system generates the\nsynthetic observational data and the forecast model in ACI at the same. This facilitates the recovery of\ncausal relationships in the absence of model error. Because the underlying system is chaotic and subject to\nstochastic forcing, a perfect model means that the model used in ACI shares the same governing equations\nand long-term statistical behavior as the system that generates the true signal. Additional results, including\nfurther numerical experiments, additional case studies, and one crucial application using sparse real-world\nobservations, are provided in the Supplementary Information.”\nOverall, the crucial role of the CIR played in understanding intermittent extreme events (Sections 3.1 and\nSI.4.1), random cause–effect role switches (Section SI.4.2), and conditional causal relationships in realistic\nmodelsofcomplexEarthprocesses(Sections3.2andSI.4.3)collectivelyprovidesupportiveevidenceofACI’s\nabilitytocapturetemporalcausalinfluences,astheframework’soutputsalignwellwithestablishedphysical\nunderstandings. This is further illustrated in the revised Supplementary Figure 4.2, which clarifies the\nunderlying mechanisms and the interplay between the subjective and objective CIRs (see point also (4) at\nthe end). In addition, the newly added case study using real ENSO data (Section 4.3.3 in the revised SI)\noffers further validation, thanks to the reviewer’s helpful suggestion.\nThemanuscript’sclaimofscalabilitytohigh-dimensionalsystemsisnotsupportedbytheoreticalanalysisor\nempiricalevidence. Tosubstantiatethisclaim,theauthorsshouldeitherprovideadiscussionofthemethod’s\ncomputationalcomplexityasafunctionofthenumberofvariables,ordemonstrateitspracticalperformance\non higher-dimensional datasets. Currently, all experiments are limited to systems with at most six state\nvariables, which does not convincingly demonstrate scalability.\n▶ Authors’ Response: We thank the reviewer for raising this question. ACI exploits Bayesian data assimi-\nlation, which allows it to leverage the computationally efficient algorithms and techniques developed in this\nmature field. These tools enable the method to efficiently perform the required computations and store\nauxiliary components (e.g., filter covariances, gain operators). In other words, scalability is supported by\nthe broad suite of computational strategies already established in Bayesian data assimilation.\nPart of this is demonstrated through the use of the adaptive-lag online smoother for CGNSs, which is\nemployedinthisworktoefficientlyestimatetheobjectiveCIRsateachtimestep(seeSections2.2–2.3ofthe\nSI). Yet, since the technique details are in the SI, we have summarized this point in the revised main text.\nWhen ensemble-based methods are used, the goal is typically to obtain an approximate representation of\nthe solution rather than the exact distribution. In practice, the ensemble size (often on the order of O(100))\nis chosen largely independently of the system’s dimension. The associated approximation error is, of course,\nanimportanttopic, butitisdistinctfromthepresentstudyandwarrantsaseparatethoroughinvestigation.\nFollowing the reviewer’s suggestion, to provide discussions on the method’s computational complexity, we\nhave added the following sentence in the introduction:\n“Third, ACI scales efficiently to high-dimensional problems by leveraging computational methods developed\nin Bayesian data assimilation. This is fundamentally different from information-transfer methods limited to\nlow dimensions.”\nWe have also added the following remarks addressing this point in the conclusion section:\n“Second, by building on the computationally efficient methods developed in Bayesian data assimilation, it\nis essential to apply ACI to higher-dimensional systems. Ensemble data assimilation is a well-established\ntechnique that routinely uses localization, covariance inflation, and other adjustments to overcome the curse\nof dimensionality [54, 55, 56]. As a result, the ensemble size (typically on the order of O(100)) is of-\nten chosen independently of the system’s dimension, which contributes to the scalability of Bayesian data\nassimilation and, consequently, of ACI. However, the sampling errors inherent in ensemble methods may\nintroduce additional biases. Thus, it is necessary to examine whether such errors affect ACI. In addition,\nvarious coarse-grained approaches, such as approximating the full distribution with a Gaussian, are widely\nused in practice. Although these approximations introduce errors in computing the PDF, they are robust and\nhighly accurate when computing diagnostic indicators, such as the relative entropy, that are used for model\nidentification and sensitivity analysis. Thus, they may substantially accelerate ACI in practice.”\nSignificance\nThis work has potentially high significance, as it addresses several major limitations of existing temporal\ncausal discovery methods. In principle, ACI can accommodate instantaneous causal effects, latent con-\nfounders,andtime-varyingcausalstructures,allfromasinglerealization. However,thesecapabilitiesremain\nlargelytheoretical, astheyarenotfullyexploredorvalidatedthroughempiricalanalysis. Additionalexperi-\nments are needed to validate the robustness of the proposed method against these factors. Additionally, the\nmanuscriptlackscomparisonswithestabelinemethods,suchascausationentropy,transferentropy,\norconvergentcrossmapping(CCM),whichlimitstheabilitytoassessACI’relativestrengthsandpractical\nutility.\n▶Authors’Response:First,wewouldliketothankthereviewerfortheirconstructiveinputandforrecogniz-\ningACI’spotentialtoaddresskeylimitationsofcurrentmethodsintemporalcausaldiscovery,particularlyits\nability to accommodate instantaneous causal effects, latent confounders, and time-varying causal structures\nfrom just a single partial realization.\nWeadditionallythankthereviewerforraisingthesehelpfulquestions. Inthiswork,weusedthedyadmodel\nwith extreme events, the predator–prey model, and a nonlinear model for the El Nin˜o–Southern Oscillation\n(ENSO)inaperfect-modelsettingtostudyinstantaneouscausaleffects,latentconfounders,andtime-varying\ncausalstructures,allfromasinglerealization. Theunderlyingdynamicalpropertiesofthesemodelsarewell\nunderstood,whichallowsustousethemtovalidatetheresultsproducedbyACI.Themethodshowedstrong\nconsistencyacrossmultiplechallengingtopics,includingextremeevents,intermittency,regimeswitching,and\nmultiscale interactions. Because ACI relies on only one time series, meaning we have only one data point\nat each time instant, knowledge of the underlying dynamics is crucial for these validation tests. Moreover,\nin all examples we used a perfect-model setup and a nonlinear data assimilation algorithm with closed-form\nanalyticexpressions. Asaresult,thereisnomodelerrorandnosamplingerror(aswouldarisewithensemble\nmethods),andthereforeanempiricalanalysisofsucherrorsourcesisnotneededhere. Wewishtoemphasize\nthat this paper focuses on the methodology and the rigorous theoretical justification of the ACI framework.\nThe specifically designed nonlinear and non-Gaussian test cases further illustrate rich features of complex\nnatural phenomena. Practical issues, many of which depend on the application, will be addressed in several\nfollow-up papers, as outlined above and in the discussion section.\nFollowingthereviewer’ssuggestion,wehavealsoaddedanadditionalexampleintheSIthatusesrealENSO\ndata to validate the method (Section 4.3.3 in the revised SI). This case is no longer a perfect-model test.\nThus, it provides a useful demonstration of the robustness of the method and offers initial evidence that\nACI remains valid under realistic observational conditions (noisy and sparse/limited data) and tolerates a\nreasonable amount of model error.\nSince ACI focuses on instantaneous causal detection and the calculation of instantaneous causal inference\nranges, it addresses different aspects of causality compared to causation entropy, transfer entropy, or CCM.\nThese are excellent methods with many valuable applications, but they primarily characterize overall or\ntime-aggregated causal relationships. Consequently, they are not suitable benchmarks for validating ACI’s\ninstantaneous causal detection or instantaneous causal ranges, and we therefore do not perform direct com-\nparisons. We appreciate the reviewer raising this point and have clarified this distinction in the discussion\nsections, which provides a qualitative comparison of the intended uses of these methods:\n“Finally, ACI requires only a single time series but assumes access to a dynamical model, which enables it to\nidentifyinstantaneouscausalrelationships. Incontrast,mosttime-series-basedmethodsoperateundermodel-\nfree assumptions and therefore focus on aggregated or time-averaged causal links rather than on causality at\nspecific moments. Because ACI incorporates model dynamics, it does not require long time series to detect\ncauses and effects. Instead, it can infer instantaneous causal links from a single data point combined with\nmodel information. In contrast, most purely data-driven methods rely on long time series and can only reveal\ncausal relationships in a time-averaged sense.”\nWe note that a potential, but limited, form of comparison in this direction would be to time-average the\nACI or CIR outputs to obtain empirical indicators. However, such averaging removes the unique instanta-\nneous information that ACI is designed to capture. Moreover, the metrics used by the different methods are\nfundamentally different, which may further complicate any meaningful comparison.\nData and methodology\nAll experiments are conducted on low-dimensional, synthetic datasets generated by the authors, including\nbenchmark systems such as the nonlinear Dyad model and a simplified ENSO model. While these exam-\nples effectively demonstrate the method’s core ideas, they offer limited insight into ACI’s performance on\nreal-world data, which typically involve higher dimensionality, observation noise, and potential violations\nof the CGNS assumptions. As no real-world datasets are analyzed, and important factors like sensitivity\nto noise, model mismatch, or partial observability are not systematically examined, it is difficult to as-\nsess the method’s robustness and generalizability. Including an experiment on a real-world dataset would\nsubstantially strengthen the methodological contribution.\n▶ Authors’ Response: Thank you for bringing this up. Our goal with this paper is to introduce the ACI\nframework and demonstrate its effectiveness, performance, and robustness across a range of demanding\nscenarios, including intermittent extreme events, causal-role reversals, and complex real-world geophysical\nphenomena. Since the methodology already involves many crucial points and the results include interesting\nfindings, we wish to focus on clarifying these points. The framework will eventually be applied to more\nrealistic systems, along the lines of what the reviewer points out which we sincerely appreciate. However,\nas we responded previously, extending the framework to the entire set of practical cases requires substantial\nadditional work and careful considerations, including addressing model error, sampling biases, and other\nissues the reviewers pointed out. This may deserve another two or three entire papers to study. We have\nadded many discussions of these points, as the reviewer suggested.\nWealsoacknowledgethereviewer’ssuggestionandhavethereforeincludedanexperimentonreal-worlddata,\nwhich will strengthen ACI’s methodological contribution by accounting for observations that are impartial,\nnoisy,incomplete/irregular,observedatdiscreteintervals,andnotnecessarilystemmingfromtheunderlying\ndynamical model. Following the reviewer’s suggestion, we have added a new subsection to the SI of this\nwork, Section 4.3.3 in the revision: “ACI on Real-World ENSO Data”, where we apply the conditional ACI\nframework on the conceptual ENSO model in (4.3) with observations defined by daily (for τ) and monthly\n(for all other variables) data for the state variables. The observations come from the following reanalysis\ndatasets (instead of using model-generated values):\n1. NCEP Global Ocean Data Assimilation System (GODAS)\n2. NCEP-NCAR Reanalysis 1 Project\nWe have also updated our“Data Availability Statement”in the main text accordingly. The details can be\nfound in this new subsection in the revised SI. We have also summarized the key findings at the end of the\ncorresponding results section in the main text:\n“In addition, in the Supplementary Information, a case study using real-world ENSO observations is exam-\nined, which does not assume a perfect model setup. The conclusions derived about the causal relationships\nusing real-world observational data remain qualitatively similar to those with model-generated data. The\nresults provide preliminary evidence that the ACI framework remains reasonably robust under realistic con-\nditions, including observational noise, moderate model error, and numerical discretization effects.”\nAnalytical approach\nThe analytical framework proposed in the paper is creative and well thought out. The derivation of estima-\ntors under the Conditional Gaussian Nonlinear System (CGNS) assumption is detailed and appears sound,\nthough I did not verify all low-level derivations. The use of infinite uncertainty for the non-target variables\n(i.e., x ) is a subtle but intuitive modeling choice, and the justification provided aligns well with Bayesian\nB\ndata assimilation principles. However, the framework’s heavy reliance on the CGNS assumption needs to\nbe examined in more detail. Since the tractability of ACI hinges on this assumption, it would be helpful\nto understand how critical it truly is. How does performance degrade when the system violates this condi-\ntion? An empirical test of the method under such violations would strengthen the argument for its broader\napplicability.\n▶ Authors’ Response: We would like to thank the reviewer for the positive and encouraging comments\nregarding our proposed methodology for assessing unconditional and conditional instantaneous causal rela-\ntionships. We also thank the reviewer for pointing out this issue, as readers may otherwise misunderstand\nthe applicability of the ACI method. Although the test models used in this work belong to the CGNS class,\nthe ACI framework itself is designed for general nonlinear dynamical systems. As explained previously, we\nuse CGNSs in the test cases to create a perfect-model scenarios with closed-form analytical expressions,\ntherebyavoidingpotentialnumericalerrorsandallowingustofocusentirelyontheresultsproducedbyACI\nwithout concerns about numerical biases. This choice helps clarify and validate the method. To make this\npointexplicit, wehaveaddedequations andexplanations atthebeginningofthemethodssection(Section2\ninthemaintext)andclearlyemphasizedthattheACIframeworkisdevelopedforgeneralnonlinearsystems\n(see (1)).\nAdditionally,whileACIoffersafreshperspectivebytracingcausesbackwardfromobservedeffects,itappears\nconceptually similar to transfer entropy and causation entropy, both of which also quantify the information\nflowbetweentime-laggedvariables. AdiscussioncontrastingACIwiththesemethods—particularlyinterms\nof expressivity, computational complexity, and assumptions—would help position the contribution more\nclearly within the existing literature. Without such a comparison, it remains unclear when and why ACI\nwould be preferred over these well-established alternatives.\n▶Authors’Response:Wethankthereviewerfordescribingourproposedinverse-problem-basedapproachas\na fresh perspective on causality. Because ACI reformulates classical predictive causality under Granger and\nWienerasaBayesianinverseproblemthroughdataassimilation,itdifferssubstantiallyfromotherestablished\ncausal inference approaches. This includes transfer entropy, causation entropy, and the additional methods\nexplicitly mentioned in Section 1 of the main text, such as information flow, convergent cross mapping, and\nKoopman causality. We have highlighted these distinctions in the methods sections:\n“Under (2.5), ACI provides a unique formulation for identifying causal relationships. In traditional ap-\nproaches, causal links are inferred by running (2.1) forward to quantify the informational response on the\ntarget variables’ entropy when conditioned on the history of the candidate causes. ACI instead solves an\ninverse problem, thus tracing causes back from the data of the observed subsequent effects. By interpolating\ncausality from effects to causes instead of extrapolating the latter forward-in-time, ACI offers an inherently\nmore robust framework compared to classical predictive causality.”\nWehavealsoaddedaparagraphinthediscussionsectionthatfurthercontrastsACIwiththeseothermethods\nexplicitly in terms of expressivity, computational complexity, and assumptions.\n“It is worth highlighting how ACI differs from other state-of-the-art methods in terms of expressivity, compu-\ntational complexity, and underlying assumptions. Different from most methods that identify cause-and-effect\nrelationships by running the system forward and quantifying informational responses in the target variables’\nentropy conditioned on past candidate causes, which is a predictive setting typical of information-flow-based\nmethods, ACI instead formulates causality as an inverse problem in uncertainty quantification. Within the\nBayesian data assimilation framework, candidate causal variables are inferred backward from observations\nof their potential effects. Thus, while many existing methods extrapolate causes forward in time, ACI traces\ncausesbackfromobservedeffects,effectivelyinterpolatingratherthanextrapolatingcausality. Thisdistinction\nmakes ACI more expressive in capturing instantaneous causal structures and, in practice, often more robust.\nFromacomputationalstandpoint,ACIcanleveragewell-developedBayesiandataassimilationtoolsthatscale\neffectively to high-dimensional systems [53, 36]. In contrast, information transfer and other model-based\nmethods often require computing the exact density functions and their derivatives, which is computationally\ndemanding and subject to the curse of dimensionality. Finally, ACI requires only a single time series but\nassumes access to a dynamical model, which enables it to identify instantaneous causal relationships. In con-\ntrast, mosttime-series-basedmethodsoperateundermodel-freeassumptionsandthereforefocusonaggregated\nor time-averaged causal links rather than on causality at specific moments. Because ACI incorporates model\ndynamics, it does not require long time series to detect causes and effects. Instead, it can infer instantaneous\ncausal links from a single data point combined with model information. In contrast, most purely data-driven\nmethods rely on long time series and can only reveal causal relationships in a time-averaged sense.”\nClarity and context\nWhile the overall presentation is clear, certain technical aspects would benefit from additional context and\nclarification. For example, the discussion in Section 4 regarding the use of infinite uncertainty for x\nB\nis difficult to follow in isolation. Its rationale becomes clear only when cross-referenced with Supporting\nInformation Sections 1.4.2 and 1.4.3.\n▶ Authors’ Response: Thank you for bringing up this concern. Indeed, (we assume the reviewer here is\nreferring to Section 2.3 of the manuscript) the exposition and discussion in Section 2.3 of the main text\nregarding the use of infinite uncertainty for the non-target variables x when developing the theory of\nB\nconditional ACI can be difficult to follow in isolation. Following the reviewer’s recommendation, for self-\ncontainedness and better readability, we have revised part of Section 2.3 in the main text to the following:\n“In the more general context of a complex system, state variables can be grouped into three disjoint subsets:\nx , x , and y, with x = (x ,x )T under the previous notation. The goal is to determine whether y is a\nA B A B\ncauseofx inthepresenceoftheremainingvariablesx , whicharecalledthenon-targetvariables, including\nA B\nthe cases of confounders or mediators. Since x interacts with both x and y, appropriately accounting for\nB A\nthe role of x while inferring the causal link from y to x becomes essential. Hereafter, let us assume that a\nB A\nrealization for each of x and x , namely x (s≤T) and x (s≤T), is available. Therefore, the candidate\nA B A B\ncause y is the only variable that does not require to have observations in this framework. Following the\nargument in Section 2.1, the causal relationship from y(t) to (x ,x ) is identified based on the uncertainty\nA B\nreduction in the smoother distribution ps related to the filter one pf, where\nt t\npf =p(y(t)|x (s≤t),x (s≤t)),\nt A B\nps =p(y(t)|x (s≤T),x (s≤T)).\nt A B\nThe main focus here is to appropriately determine and resolve the presence of x from pf and ps as to infer\nB t t\na conditional causal relationship from y(t) to x beyond the contributions of x .\nA B\nRecall from Section 2.1 that ACI employs Bayesian data assimilation to estimate the state of y(t). ACI\nexploits observations of x to reduce the uncertainty in y(t), where the influence of x on the estimation\nof y(t) depends on its own uncertainty: smaller observational uncertainty leads to a stronger impact\n[45]. When extending this framework to x = (x ,x )T for conditional ACI, the observed time series of x is\nA B B\nstill used in the forecast step to compute the uncertainties in x and y. However, during the\nA\nanalysis step, the uncertainty in x ’s dynamics is deliberately excluded by assigning infinite uncertainty\nB\nto its marginal likelihood before posterior computation. This can be rigorously formulated through a limit\noperation on the analysis component of the Bayesian data assimilation update equations, which formally\nleads to the following filter- and smoother-based distributions:\npf|xB := lim pf, ps|xB := lim ps.\nt t t t\nVar(xB(t))→+∞ Var(xB(t))→+∞\nSimilar to (2.5), if the relative entropy of ps|xB from pf|xB is positive, P (cid:0) ps|xB,pf|xB (cid:1) >0, then y is identified\nt t t t\nas the cause of x conditional to x at time t. This means future data of x incur additional information\nA B A\ngain about y(t) related to its history, beyond the observational contributions of x .\nB\nBy computing this metric at each t, a time-dependent conditional causal relationship is established.”\nAdditionally, the computational procedure underlying ACI is fairly complex. Including a pseudocode algorithm\nblock that outlines the major steps would improve accessibility for a broader audience and facilitate\nreproducibility.\n▶ Authors’ Response: This is an excellent suggestion! Thank you for the advice. This suggestion, as\nwell as the one of Reviewer #3 to make our codebase publicly available, will significantly improve the\naccessibility and reproducibility of this work. We have made all MATLAB m-file scripts used to reproduce\nthe analyses and figures in the main text and SI available at the following GitHub repository: https:\n//github.com/marandmath/ACI_code. To further facilitate the understanding by general audience, we also\nimplement the reviewer’s suggestion to provide pseudocode algorithm blocks for the following:\n1. Base ACI framework skeleton: See Section 1.3 in the SI (Algorithm 1).\n2. Conditional ACI method: See Section 1.4.2 in the SI (Algorithm 2).\n3. Computation of the subjective and objective CIRs, including the latter’s computationally efficient\napproximation, for CGNSs: See Section 2.3 in the SI (Algorithm 3).\nSince space is limited in the main text, all pseudocode blocks are in the SI.\nWe have chosen to include pseudocode for the computation of the CIRs specifically for CGNSs so readers\nhaveaconcreteimplementationtheycanworkwithinsteadoftheabstracttheoryfromSection1.5oftheSI.\nNevertheless, we note here that the pseudocode provided in Algorithm 3 is easily applicable to the general\nsystem in (1) by appropriately computing δ(t ;t )=Pj for each j ∈ N and n∈{j,...,N}.\nn j n\n(cid:74) (cid:75)\nSuggested improvements\n• Validation and Ground Truth Testing: Include experiments with synthetic datasets where the true\ncausal influences and CIRs are explicitly known.\n• Scalability Assessment: Provide theoretical analysis of computational complexity as a function of\nsystem dimensionality, and demonstrate ACI’s performance on higher-dimensional datasets.\n• Robustness Analysis: Conduct systematic experiments examining ACI’s sensitivity to key factors in-\ncluding observation noise, violations of CGNS assumptions, and partial observability.\n• Baseline Comparisons: Include comparative analysis with established methods such as causation en-\ntropy, transfer entropy, and convergent cross mapping (CCM).\n• Real-world Application: Add at least one experiment on real-world data to demonstrate practical\napplicability beyond synthetic examples.\n• Enhanced Clarity: Include a pseudocode algorithm block outlining ACI’s major computational steps\nto improve accessibility and reproducibility.\n▶Authors’Response:Weagainthankthereviewerfortakingthetimetoprovidetoussuchathoroughand\nconstructivereviewfullofimportantsuggestionsandfeedback. Pleaseseeourpreviousresponsestoseehow\nwe addressed each of these suggested improvements. We believe that with the implementations and changes\nmade here, we have significantly improved the quality of the manuscript.\nReferences\nThemanuscriptappropriatelyreferencesmostrelevantworksinthefield. Tostrengthentheliteraturereview,\nconsider adding a citation to Graber and Schwing (2020) [1], which specifically addresses dynamic causal\nlinks using Granger causality and provides methods for causal discovery from single realizations.\n[1]Graber,Colin,andAlexanderSchwing. “Dynamicneuralrelationalinferenceforforecastingtrajectories”.\nProceedingsoftheIEEE/CVFConferenceonComputerVisionandPatternRecognitionWorkshops. 2020.\n▶ Authors’ Response: Thank you for the positive comments and for the suggestion. We have included the\nsuggested reference in the second paragraph of the introduction (Section 1 in the main text).\nReviewer #2\nRemarks to the Authors:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNatureCommunicationsinitiativetofacilitatetraininginpeerreviewandtoprovideappropriaterecognition\nfor Early Career Researchers who co-review manuscripts.\n▶Authors’Response:Wesincerelythankthereviewerandtheirco-reviewerfortheireffortsandconstructive\nfeedback provided under the Nature Communications co-review initiative for early career researchers. Their\ninput and time is greatly appreciated.\nReviewer #3\nRemarks to the Authors:\nDear editor and authors,\nThis is my review of the manuscript entitled“Assimilative Causal Inference”.\nThe manuscript introduces, formally defines and thoroughly tests a new method for inferring causality:\nAssimilative Causal Influence (ACI). ACI promises to detect causal influence under a variety of conditions,\nincludinghigh-dimensionalsystems, evenincaseswherethestrengthanddirectionofcausalitychangesover\ntime. These conditions are often requirements in fields such as climatology, ecology and neuroscience and\npresumably also highly relevant in other areas such as physiology and economics. In addition to ACI the\npaper introduces Causal Influence Range (CIR), which is a measure for the time-scale at which the the\ncausalinfluenceiseffective. Theauthorscircumventaninherentambiguity(subjectivity)inthismeasureby\ndeveloping an objective version which is a normalized integral over all values of relative entropy.\nACI is based on inverse inference, i.e., knowing future values of the effect (at time later than t) reduces\nthe uncertainty in inferring the value of the cause at time t. This is formalised using relative entropy (KL\ndivergence) in a Bayesian data assimilation framework. The method is clearly explained in the main paper,\nwithamoredetailedexplanationintheSI.Figure2.1isalsoveryhelpfulforunderstandingACIandobjective\nCIR.\nMy overall assessment is that this is a very well-written paper and that the developed method is novel and\nwill be highly interesting and relevant to many researchers. The authors present their method and results\nin a succinct and thorough way, and my recommendation is to publish the paper more or less as is, except\nfor one suggestion and some minor issues (see below).\n▶Authors’Response:Wesincerelythankthereviewerfortheirverypositiveandencouragingassessmentof\nour manuscript. We are grateful for their recognition of the novelty, clarity, and relevance of our proposed\ncausal inference method, as well as of their thoughtful suggestion and proposed minor corrections. We have\ncarefully addressed all the points raised, making corresponding revisions and additions to the manuscript.\nWe believe these changes have further improved the clarity and overall quality of the paper and its accom-\npanying material. Our detailed responses to each comment are provided below.\nMajor comment:\nI really only have one major suggestion, which is that I think the authors should make the code used for\nanalysis and figures available up-front, instead of requiring interested readers to contact the corresponding\nauthor. I think this will be in the interest of the authors as well, since it will make the method more\naccessible. A second reason is that I have had several unfortunate experiences of writing authors to request\ncode and/or data from papers and either have not received an answer or been rejected on various grounds.\nWhile I have no reason to think that the authors of the current manuscript would not be open to sharing,\nsometime people change jobs, e-mail addresses or even lose or misplace code and data. I sincerely hope the\nauthorswillfollowthissuggestion. Thisisabeautifulandwellorganizedpaper,andIthinkhavingthecode\nwould increase its usefulness to interested readers a lot.\n▶ Authors’ Response: This is an excellent suggestion! We fully agree that open access to code enhances\ntransparency, reproducibility, and accessibility. The examples brought up by the reviewer serve as great\nreasons to why that needs to be norm for methodology papers. In response to this suggestion, we have now\nmade all code used for the analyses and the production of the figures for all of the case studies carried out\nin this work publicly available at the following GitHub repository:\nhttps://github.com/marandmath/ACI_code\nWe have also updated our“Code Availability Statement”in the main text accordingly. We have chosen to\nshare our code in MATLAB, as it strikes a good balance between accessibility and fast performance.\nFurther following the reviewer’s suggestion for wider accessibility and increased usefulness to interested\nreaders, in this repository we have made a conscious effort to meticulously document all of the code in each\nscript file with explanatory comments. We have also added a thorough docstring at the beginning of each\nm-file, both for exposition and for further implementation instructions.\nIn addition to allowing users to reproduce the analyses and figures present in this work, we have also added\ncode which generates results and experiments not present in it (due to length constraints); for example\nsubjective and objective CIR comparison plots for all case studies, and plots comparing the objective CIR\ndefinition and its computationally efficient approximation over time. We also include curated daily and\nmonthly data from the following reanalysis datasets: NCEP Global Ocean Data Assimilation System (GO-\nDAS) and NCEP-NCAR Reanalysis 1, specific to the ENSO conceptual model. These can be used by the\nuser in lieu of observations generated from the model simulation as a way to conduct a more realistic ACI\nanalysis of the ENSO diversity.\nWe believe that with this addition we have increased the usefulness of our work to a wide range of practi-\ntioners and theorists. We again thank the reviewer for this very useful suggestion, as well as for their very\nflattering comments regarding our manuscript.\nMinor comments:\n• In the abstract the authors write that ACI“provides online tracking...” I was initially confused about\nthe use of the word online and prefer the formulation they use on page 1, line 39: “real-time tracking\nof the causal relationship”which, if I understand correctly, means the same thing.\n▶Authors’Response:Indeed,wehaveused“real-time”heretoreferto“online”,asinouruseof“online”\nin Section 2.2 of the SI regarding the online smoother for CGNSs, to denote the real-time smoother\nalgorithminthescenariowhereobservationsareobtainedsequentiallyovertime. Overall,“online”isa\nmore precise term. Therefore, we have revised the abstract and changed“real-time”with“online”, per\nthe reviewer’s suggestion. We have done the same for another similar instance of the use of“real-time”\nfrom the introduction.\n• On p. 5, l. 173 the wrong symbol is used. It should be ε, not ϵ for the threshold parameter.\n• On p. 7, l. 274 the same mistake recurs.\n▶ Authors’ Response: Thank you so much for catching these! We have replaced both instances of ϵ\n(\\epsilon) with ε (\\varepsilon), including two more instances that happened in the SI.\n• In section 2.3 the variable x is split into two subsets x and x , but this is not mentioned explicitly\nA B\n(except in the SI). It would be helpful to state briefly in the main text as well, since x is used later on\nin the section to refer to the union of x and x (e.g. on p. 6, l. 234–235).\nA B\n▶ Authors’ Response: Thank you for bringing this to our attention. We have now clarified in the\nrevision of Section 2.3 in the main text that x=(x ,x )T. Here is the revised excerpt: “In the more\nA B\ngeneral context of a complex system, state variables can be grouped into three disjoint subsets: x ,\nA\nx , and y, with x=(x ,x )T under the previous notation.”.\nB A B\n• Insection3.1,Eq. (3.1a)and(3.1b): W andW (andtheirderivatives)arenotdefinedorexplained.\nx y\nAgain the SI will help the reader, but a short comment here would help.\n▶ Authors’ Response: Thank you for bringing this up. Indeed, we have not concretely defined Wiener\nprocesses in the main text for brevity. We have added the following exposition to the beginning of\nSection2.1.1inthemaintextwhichbrieflydefinesandexplainsthisconcepttothereaderindependent\nof the SI.\n“Consider a time interval [0,T] and a specific time instant t, where 0 ≤ t ≤ T. Let {x(t)} and\n0≤t≤T\n{y(t)} be two multivariate stochastic processes defined over this interval. Denote by x(t) and\n0≤t≤T\ny(t) the corresponding multivariate random variables at time t. Their evolution in time is defined by\nthe following stochastic dynamical system:\ndx\n=fx(t,x,y)+Σx(t,x)W˙ +Σx(t,x)W˙ ,\ndt 1 1 2 2\n(2.1)\ndy\n=fy(t,x,y)+Σy(t,x,y)W˙ +Σy(t,x,y)W˙ ,\ndt 1 1 2 2\nwhere W˙ and W˙ are independent mean-zero Gaussian random vectors with uncorrelated components\n1 2\nthat possess unit variance. Mathematical details regarding the structure of (2.1) are given in the\nSupplementary Information.”\n• Figure 3.1: If space permits, it would be interesting to see a third panel with a plot of objective CIR\nas a function of t. I find it a bit difficult to decode the whisker plot in panel (a).\n▶ Authors’ Response: As a way to succinctly convey the concept of the forward- or future-pointing\nobjective CIR, while minimizing the length of the main text, we have decided to use that whisker plot\nconvention in Figure 3.1 to denote the forward CIR lengths for y(t)→x at each time t going forward\nfrom y(t).\nWeagreewiththereviewerthatthesemightbesomewhatdifficulttodiscernordecodeonsomescreens\norinprintedformat. Assuch,wehavedecidedtoaddanadditionalpanel,Panel(c),toSupplementary\nFigure 4.2 in the SI (Panel (c) from before,“ACI Metric for y(t)→x”, is now Panel (d); see also point\n(4) at the end). This newly added plot serves two purposes:\n1. Clearly showcases the objective CIR as a function of t for the time window corresponding to\nFigure 3.1 in the main text. This is exactly what the reviewer suggested.\n2. Additionally showcases the skill of the computationally efficient approximation we have adopted\nin this work to effectively estimate the objective CIR lengths without resorting to the much\nmorecomputationallyintensiveε-integralsinvolvedinthedefinition. Specifically,itcomparesthe\napproximate objective CIR with the exact objective CIR over time t, where each exact CIR was\nevaluated using a quadrature of the corresponding subjective CIRs over ε.\nWehopethatwiththisadditionwehavenotjustaddressedthereviewer’sconcerns,butalsoilluminated\nfurther the numerics behind the objective CIR, specifically the efficacious recovery of the objective\ntemporal measure of causal impact through its computationally efficient approximation.\nRegards,\nDan Mønster\nReviewer #4\nRemarks to the Authors:\nReview: Assimilative Causal Inference, by Andreou, Chen and Bollt\n▶ Authors’ Response: We sincerely thank the reviewer for taking the time to provide a thorough and con-\nstructive review of our paper. In what follows, we carefully outline how each of the concerns raised and\nsuggestions given by the reviewer have been addressed and adopted in the revised manuscript. We believe\nthese changes have substantially improved the clarity, rigor, and overall quality of the paper, and for that\nwe again thank the reviewer.\nThe authors introduce a framework for Assimilative Causal Inference (ACI), which harnesses Bayesian data\nassimilation to trace the candidate causes of observations. It is demonstrated to be effective for high-\ndimensional dynamical systems and short datasets, providing real-time tracking of causal roles and estab-\nlishing a mathematically rigorous criterion for the causal influence range.\nThe authors describe the general significance of ACI across various scientific disciplines and some of the\nlimitations of existing methods in capturing instantaneous, time-evolving causal relationships in complex\nsystems. Theapproachisbothpowerfulinscopeofapplicationandrigorousinaself-consistentsenseandthus\nshouldbeofinteresttoabroadscopeofresearchersinterestedincomplexdynamicalsystems. Inmyopinion\nthe paper should be published following a more sober assessment of its limitations and comparison/contrast\nto other approaches, such as those described in the second paragraph of the introduction.\n▶ Authors’ Response: We thank the reviewer for their thoughtful assessment of our work. We greatly\nappreciate their positive feedback regarding the effectiveness, self-consistency, and rigor of our approach, as\nwell as its potential to be of interest to a broad scope of researchers in complex dynamical systems.\nFor example, I find it intellectually and emotionally distasteful for an author to claim their approach is\n“paradigm-shifting”, as is done in the second sentence of the abstract, upon its introduction? Paradigm\nshifts require a broad adoption and consensus in the community and years of testing and refinement. This\nconcept was introduced by Thomas Kuhn in his “The Structure of Scientific Revolutions”, Univ Chicago\nPress, 1962. Paradigm shifts in science are not defined by fiat by an author and they take time to evolve.\nHow do the authors know that next week, month or year, etc. another method will not appear that is far\nsuperior and supersede all others? Indeed, let the community and time decide what is paradigm shifting.\n▶ Authors’ Response: We appreciate the reviewer’s thoughtful comment regarding our use of the phrase\n“paradigm-shifting”intheabstract. Wefullyagreethatsuchterminologyshouldbereservedfordevelopments\nthat have undergone broad community adoption and long-term validation, consistent with Kuhn’s original\nnotion. We sincerely apologize for the misuse of this phrase, partially due to the fact that the first two\nauthors are not English native speakers. Our use of the phrase was indeed inappropriate, and we have\nremoved it from the revised manuscript. Thank you again for pointing this out, which helps avoid many\npotential issues.\nThe paper highlights the potential of ACI to open new avenues for studying complex systems with tran-\nsient causal structures and suggests further areas for research and application of ACI. It is one important\nmethod of identifying and tracing back causes from observed effects through Bayesian data assimilation\nand can identify interactions without direct candidate cause observations. It can provide real-time tracking\nand mathematically validated causal influence range and was shown to be effective in systems showcasing\nintermittency and extreme events.\n▶Authors’Response:Weagainthankthereviewerfortheirpositivecommentsandencouragingassessment\nofACI’spotentialasanewmethod,notjustfortemporalcausalinference,butalsoforprobingintocomplex\nsystems through its formulation of the causal influence range. We also appreciate their recognition of how\nACIcanconductcausalinferencenaturally,withoutrequiringdirectobservationsfromthecandidatecauses.\nAs is noted, there are both general limitations and application-specific limitations, that make the paradigm-\nshifting claims specious. These include:\n(1) ThefactthattheACIframeworkmaybeaffectedbythecorrelationbetweenvariablesduringBayesian\ndata assimilation, which can potentially introduce a spurious causal relationship, even if conditioned\nover intermediary variables. This can create erroneous interpretations of data.\n▶Authors’Response:Thankyouforraisingthisimportantquestion. Letusfurtherclarifythemecha-\nnismsbehindtheconditionalACIframeworkdevelopedinthiswork,asitprovidesasystematicmethod\nto resolve spurious correlations and associations when assessing conditional causal relationships. See\nSection 2.3 of the main text and Section 1.4 of the SI. Conditional ACI differs fundamentally from ap-\nproaches that condition or marginalize over the (lagged) history of intermediary or ancillary variables\nthat may act as confounders, mediators, moderators, and more. In this sense, it stands in contrast to\ntraditional causal inference approaches such as transfer entropy and causation entropy, as discussed in\nSection 2.3 and further elaborated in Sections 1.4.3–1.4.4 of the SI.\nUnlike many state-of-the-art methods that rely on empirical procedures to remove spurious associa-\ntions, conditional ACI addresses this issue naturally and self-consistently through its inverse-problem\nformulation of causality using Bayesian data assimilation. The core ACI structure combines model\nforecasts with observational information from potential subsequent effects to identify their candidate\ncauses. In conditional ACI, the non-target variables x continue to influence the underlying system\nB\ndynamics through the filtering and smoothing equations. However, their observational influence on\nthe estimation of the unobserved candidate causes is explicitly removed. This allows the method to\ntrace the causality of the observed effects back to their genuine causes. Importantly, this procedure\npreserves the structure and integrity of the governing dynamical equations, distinguishing conditional\nACI from many other model-based causal inference techniques.\nRegarding the correlations mentioned by the reviewer during the state estimation of y(t), specifically\n(a) between x and y(t), and (b) between x and x , the conditional ACI formulation is deliberately\nB B A\nconstructedtoremovetheseeffects. Bylettingthemarginallikelihooduncertaintyofx tendtoinfinity\nB\nduring the analysis step, formally Var(x (t)) → +∞, the corresponding correlations are effectively\nB\nignoredorresolvedwheninferringtheconditionalcausalrelationship(y(t)→x )|x . Theresolution\nA B\nof (a) through this mechanism has been discussed in Section 2.3 of the main text and Section 1.4 of\ntheSI.Thankstothereviewer’ssuggestion, wehaveaddedmorediscussionsinSection2.3ofthemain\ntext to clarify this argument. We provide the relevant excerpt here:\n“In the more general context of a complex system, state variables can be grouped into three disjoint\nsubsets: x , x , and y, with x = (x ,x )T under the previous notation. The goal is to determine\nA B A B\nwhether y is a cause of x in the presence of the remaining variables x , which are called the non-\nA B\ntarget variables, including the cases of confounders or mediators. Since x interacts with both x and\nB A\ny, appropriately accounting for the role of x while inferring the causal link from y to x becomes\nB A\nessential. Hereafter, let us assume that a realization for each of x and x , namely x (s ≤ T) and\nA B A\nx (s ≤ T), is available. Therefore, the candidate cause y is the only variable that does not require to\nB\nhave observations in this framework. Following the argument in Section 2.1, the causal relationship\nfrom y(t) to (x ,x ) is identified based on the uncertainty reduction in the smoother distribution ps\nA B t\nrelated to the filter one pf, where\nt\npf =p(y(t)|x (s≤t),x (s≤t)),\nt A B\nps =p(y(t)|x (s≤T),x (s≤T)).\nt A B\nThe main focus here is to appropriately determine and resolve the presence of x from pf and ps as to\nB t t\ninfer a conditional causal relationship from y(t) to x beyond the contributions of x .\nA B\nRecallfromSection2.1thatACIemploysBayesiandataassimilationtoestimatethestateofy(t). ACI\nexploits observations of x to reduce the uncertainty in y(t), where the influence of x on the estimation\nof y(t) depends on its own uncertainty: smaller observational uncertainty leads to a stronger impact\n[45]. When extending this framework to x = (x ,x )T for conditional ACI, the observed time series\nA B\nof x is still used in the forecast step to compute the uncertainties in x and y. However, during the\nB A\nanalysisstep, theuncertaintyinx ’sdynamicsisdeliberatelyexcludedbyassigninginfiniteuncertainty\nB\ntoitsmarginallikelihoodbeforeposteriorcomputation. Thiscanberigorouslyformulatedthroughalimit\noperationontheanalysiscomponentoftheBayesiandataassimilationupdateequations,whichformally\nleads to the following filter- and smoother-based distributions:\npf|xB := lim pf, ps|xB := lim ps.\nt t t t\nVar(xB(t))→+∞ Var(xB(t))→+∞\nSimilar to (2.5), if the relative entropy of ps|xB from pf|xB is positive, P (cid:0) ps|xB,pf|xB (cid:1) > 0, then y is\nt t t t\nidentified as the cause of x conditional to x at time t. This means future data of x incur additional\nA B A\ninformation gain about y(t) related to its history, beyond the observational contributions of x .\nB\nBy computing this metric at each t, a time-dependent conditional causal relationship is established.”\nFor (b), the details are somewhat more subtle, and we acknowledge that the SI may not have fully\nelaborated on this point. While we alluded to the handling of correlations in the observed state space\nduringtheproofoftheprincipleofnilconditionalassimilativecausalityforCGNSsinSection3.2ofthe\nSI, we did not restate this explicitly in Section 1.4. In essence, by sending Var(x (t)) → +∞ during\nB\nanalysis, the correlations between x and x are also driven to zero at that time t. Consequently,\nA B\nobservational correlations are simultaneously resolved during the state estimation of y(t), ensuring\nthat within the ACI framework the observed subsequent effects x are traced back to their genuine\nA\ncandidatecausesy(t). Inthisregard,conditionalACIcanbelooselycomparedtoassigninganimproper\nor diffuse prior in parametric Bayesian inference. By assigning infinite uncertainty to the marginal\nlikelihood of x during the analysis step, x effectively“uncorrelates”from x , thereby allowing only\nB B A\nthe future information of x to inform the state estimation of y(t) beyond its own history. Following\nA\nthe reviewer’s concern, we briefly summarize the discussion regarding (b) from above and include it in\nthe following revision of point (1) from Section 1.4.3 in the SI.\n“Complete elimination of x ’s influence during data assimilation, ensuring any uncertainty reduction\nB\nin y(t) stems solely from x , consistent with our ACI objectives. Furthermore, during analysis, the\nA\ncorrelations between x and x in their likelihood are necessarily sent to zero as the uncertainty of x\nA B B\ngoes to infinity. This further resolves the potential for spurious associations arising over the observable\nstate space that can flow over when inferring causality from y(t) to x ; see Theorem 3.2. A rough\nA\nanalogue to this approach would be the assignment of an improper prior as an uninformative or diffuse\nprior in parametric Bayesian inference.”\n(2) Thereal-timetrackingcapabilityoftheACIframeworknecessitatescontinuousobservationofadataset.\nThis requirement can be challenging in systems where data is not continuously available or is incom-\nplete, which is often the case.\n▶ Authors’ Response: Thank you for bringing this important question up. Actually, we wrote in\nSection2.1.1ofthemaintext,“For simplicity, the time series are assumed to be continuously observed.\nAn analogous framework can be developed for discrete-in-time observations”, and in Section 1.1 of the\nSI,“For convenience, the time series of x is assumed to be continuously observed, with the development\nof an analogous framework for discrete-in-time observations being possible”. The procedure of the ACI\nframework can be naturally adopted to the cases where the observations of x are obtained at discrete\ntime intervals. The same applies to the assumption of non-uniform or non-constant observational\nrates, which can lead to incomplete datasets. We acknowledge that this point may not have been\nsufficientlyemphasizedintheoriginalversion. Inresponsetothereviewer’ssuggestion, wehaveadded\nthe following paragraph in the discussion section of the main text to highlight this aspect:\n“Finally, continuous-in-time observations are adopted in this work for simplicity of presentation. Nev-\nertheless, an analogous framework can be naturally extended to discrete-in-time observations using the\ncorresponding Bayesian data assimilation algorithms. An interesting scientific question in this con-\ntext is to understand the uncertainty in causal detection arising from the discrete nature of the data,\nparticularly at time intervals between successive observations. This line of inquiry could also highlight\nan important distinction between ACI and purely data-driven methods. In ACI, the dynamical model\nprovides a natural way to interpolate causal relationships at time instants when observations are un-\navailable. Therefore, it may help bridge observational gaps and maintain temporal continuity in causal\ninference.”\nWe have also implemented a new test using real-world observations that are noisier and which also\nintroduces model and discretization errors into the framework. This is in Section 4.3.3 of the SI.\nWe\nhave added the following at the end of the results section:\n“In addition, in the Supplementary Information, a case study using real-world ENSO observations is\nexamined, which does not assume a perfect model setup. The conclusions derived about the causal rela-\ntionships using real-world observational data remain qualitatively similar to those with model-generated\ndata. TheresultsprovidepreliminaryevidencethattheACIframeworkremainsreasonablyrobustunder\nrealistic conditions, including observational noise, moderate model error, and numerical discretization\neffects.”\nOne supporting evidence toward a discrete-in-time formulation in this work is illustrated by the use of\nthe discrete-in-time online smoother for CGNSs to analytically compute the CIRs in the case studies\n(see Sections 2.2–2.3 of the SI). This algorithm can accommodate x observed at discrete, possibly\nnon-uniform,intervals. Nevertheless,theuseofdiscreteobservationsforxcanalsobenaturallyimple-\nmented in ACI through consistent discretizations of the posterior filter and smoother state estimation\nequations. For the general stochastic complex systems assumed in this work (see (2.1) in the revised\nmain text and (1.1) in the SI), these ideas naturally extend to that case too. Specifically, we can\nleverage the powerful results of Bayesian data assimilation concerning the filter and smoother state\nestimation equations that track the evolution of the associated PDFs through exact random PDEs\nwhich we provide in their unnormalized form in (1.8) and (1.11) in the SI, respectively. In the case\nwhere x is observed at discrete time intervals, these equations (or any of their approximations, e.g.,\nensemble variations) can be discretized in time through an appropriate numerical model, with the\ncontinuous-in-time ACI formulation giving rise to a discete one completely analogously. The same\nresultalsoappliestotheCIRanalysisthroughappropriatelydiscretizedonlinesmootherformulations,\nas already described in Section 2.3 for CGNSs.\n(3) AlthoughACIclaimstoscaleefficientlytohighdimensions,thecomplexityoftheframeworkstillposes\nsignificant computational challenges, particularly in very large-scale systems.\n▶Authors’Response:Thankyouforraisingthisimportantquestion. Asdiscussedintheintroduction,\nsince ACI is built upon Bayesian data assimilation, it naturally benefits from the many numerical\nmethods already developed in that field to handle high-dimensional systems. Therefore, ACI can, in\nprinciple, be scaled to address such problems as well. The primary goal of this paper is to develop the\nACI framework and to distinguish it from existing methods, both in terms of its methodology and the\nnew features it reveals. For this reason, we have focused on relatively lower-dimensional systems for\nnumerical illustrations. Extending the framework to much higher-dimensional systems is conceptually\nstraightforward, although careful numerical design is required, which is an effort that could merit one\nor more separate studies for full implementation. Nevertheless, we appreciate the reviewer’s insightful\ncomment and have added a paragraph in the discussion section outlining a more concrete path toward\napplying ACI to high-dimensional systems.\n“Second, by building on the computationally efficient methods developed in Bayesian data assimilation,\nit is essential to apply ACI to higher-dimensional systems. Ensemble data assimilation is a well-\nestablished technique that routinely uses localization, covariance inflation, and other adjustments to\novercome the curse of dimensionality [54, 55, 56]. As a result, the ensemble size (typically on the\norder of O(100)) is often chosen independently of the system’s dimension, which contributes to the\nscalability of Bayesian data assimilation and, consequently, of ACI. However, the sampling errors\ninherentinensemblemethodsmayintroduceadditionalbiases. Thus,itisnecessarytoexaminewhether\nsuch errors affect ACI. In addition, various coarse-grained approaches, such as approximating the full\ndistribution with a Gaussian, are widely used in practice. Although these approximations introduce\nerrorsincomputingthePDF,theyarerobustandhighlyaccuratewhencomputingdiagnosticindicators,\nsuch as the relative entropy, that are used for model identification and sensitivity analysis. Thus, they\nmay substantially accelerate ACI in practice.”\n(4) The framework assumes the underlying models are correctly specified without observational noise, a\ncondition that is rarely true in real-world applications where measurements are often noisy in many\ndistinct, but difficult to quantify, ways.\n▶ Authors’ Response: Thank you for bringing up these valid concerns. These are indeed crucial prac-\ntical issues for all causal inference methods. While detailed investigations would warrant one or more\ndedicated papers, we appreciate the reviewer’s insightful comments and have expanded the discussion\nalong these lines in the revised manuscript. First, the Bayesian data assimilation framework underly-\ning ACI does not require a perfect model. Thus, a misspecified model can be naturally incorporated\nwithinACItostudyitseffects. Byperformingtwoparallelexperiments(oneusingaperfectmodeland\nanother using an imperfect model), we can directly assess the impact of model error on causal infer-\nence. More importantly, based on the reviewer’s insights, ACI itself can be leveraged to detect model\nerror and to help construct parsimonious, data-informed dynamical models. Although this extends\nbeyond the core goal of causal inference, it represents a broader contribution to model identification\nand the dynamical systems community. Second, because ACI is formulated within the Bayesian data\nassimilation framework, it can also address the influence of observational noise. A natural strategy\nis to apply a pre-filtering step to mitigate part of the observational noise before conducting ACI. An\neven more informative study is to compare the ACI results obtained from the original noisy data with\nthose from the pre-filtered data. The differences between the two provide a valuable diagnostic of how\nobservationalnoiseaffectscausalinference. SinceBayesiandataassimilationisalreadyintegraltoACI,\nsuchpre-filteringcanbeperformedusingmodel-basedinformationratherthanempiricalnoise-removal\nmethods, as required in many other approaches. Overall, these directions represent important and\npromising topics that merit careful and systematic investigation in future studies. We have added the\nfollowingparagraphinthediscussionsectiontosummarizetheideasofusingACItostudythesecrucial\ntopics:\n“First, because model error is ubiquitous in practice, it is essential to study its impact on causal infer-\nence. Understanding how model error affects inferred causal relationships can create opportunities not\nonly to detect and correct model error, but also to leverage this information to develop parsimonious\nmodels for complex dynamical systems. Similarly, observational data may contain additional noise that\ncan affect causal inference. The ACI framework provides a natural way to analyze how observational\nnoise contaminates causal detection. In such a context, data assimilation can serve as a pre-filtering\nstep to reduce the influence of observational noise, which allows for a systematic examination of its\nrole in shaping causal inference outcomes.”\n(5) While the authors noted the applicability in short datasets, the accuracy and reliability of causal\ninferences drawn from very limited data remain a concern.\n▶ Authors’ Response: We thank the reviewer for raising this crucial question. Short datasets indeed\npose significant challenges for most data-driven causal inference methods. However, because ACI\nintegrates model information with observational data, it requires only a single data point at each\ntime instant to identify instantaneous causal relationships. This represents one of the fundamental\ndistinctions between ACI and standard data-driven approaches. We thank reviewer again for this\ncomment,whichhelpsusimproveourpresentation. Tothisend, wehaveclarifiedthisdifferenceinthe\ndiscussion section:\n“Finally, ACI requiresonlya singletimeseriesbut assumesaccessto adynamicalmodel, which enables\nit to identify instantaneous causal relationships. In contrast, most time-series-based methods operate\nunder model-free assumptions and therefore focus on aggregated or time-averaged causal links rather\nthan on causality at specific moments. Because ACI incorporates model dynamics, it does not require\nlong time series to detect causes and effects. Instead, it can infer instantaneous causal links from a\nsingle data point combined with model information. In contrast, most purely data-driven methods rely\non long time series and can only reveal causal relationships in a time-averaged sense.”\n(6) Theapplication-specificlimitationsinclude(a)extremeeventanalysis,whereaccurateparameterselec-\ntion and model calibration are essential; (b) historical data limitations, such as shifts not represented\nin a dataset and (c) interference handling, which can compromise the reliability of the inferred causal\nrelationships.\n▶Authors’Response:Wethankthereviewerforraisingthisthoughtfulquestion. Ifourunderstanding\nis correct, the reviewer is referring to the accuracy of the model used in the extreme event study and,\ningeneral,theACIframework. Indeed,thisaccuracyisessential. Inouranalysisofextremeevents,we\nemployed a perfect-model setup, which we have clarified in the revised version of the main text. The\nterm perfect model does notimplythat the model reproduces the exact trajectory of the observations.\nBecause the underlying system is chaotic and subject to stochastic forcing, a perfect model means\nthat the model used in ACI shares the same governing equations and long-term statistical behavior as\nthe system that generates the true signal. Accurate parameter estimation and model calibration are\ntherefore crucial. The historical data used for training should not be statistically biased, although it\nis not necessary for the exact events in the testing period to appear in the training dataset. Together\nwith the response to question (4) of the reviewer, we have expanded the discussion section to include\nan explicit consideration of model error and its implications for ACI:\n“First, because model error is ubiquitous in practice, it is essential to study its impact on causal\ninference. Understandinghowmodelerroraffectsinferredcausalrelationshipscancreateopportunities\nnotonlytodetectandcorrectmodelerror,butalsotoleveragethisinformationtodevelopparsimonious\nmodels for complex dynamical systems.”\nWhiletheselimitationsarenoted, andnotable, thereaderwillstronglybenefitfromknowinghowthesefew,\nof potentially many more, are treated in other approaches in the field, many of which are mentioned in the\nintroduction and are known to the authors in more detail.\n▶ Authors’ Response: We again thank the reviewer for taking the time to provide this constructive review\nwith crucial feedback, and valid and important concerns. We have added discussions and clarifications to\nexplain the issues raised by the reviewer. With the implementations and changes made here based on the\nreviewer’s suggestions, we wish we have improved the quality of the manuscript.\nWe sincerely thank all of the reviewers for their time and suggestions!\nAdditional changes independent of the Reviewers’ comments:\n(1) In the figures corresponding to the ENSO case study (Figure 3.2 in the main text and Supplementary\nFigures4.5–4.8intheSI),eachoftheENSOeventcolorlegendsonthey-axis(orx-axisaccordingly)has\nbeen changed to now range from September to February, which coincides with the peak of the actual\nevent during boreal winter. Furthermore, we have slightly increased the threshold for defining ex-\ntremeEPElNin˜oevents(increasedby0.5◦CfortheaverageduringborealwinterDecember–January–\nFebruary),astocaptureactualsignificantextremeeventsandnotmarginallystrongermoderateEPEl\\Nin˜oevents. Ofcourse,thesechangesdonotalteranyofthenumericspresentintheinitialsubmission,\nsincetheyarepurelycosmeticinnature. Thegoalofthesechangesistoimprovefigurereadabilityand\ninterpretability.\n(2) MinorchangeshavebeenmadeinPanels(f)–(g)ofFigure3.2inthemaintextandtheSupplementary\nFigures4.5–4.6intheSI,whichleadtoonlyaslightchangetothetemporalprofileoftheACImeasure,\nwhich do not alter or change any of the conclusions made in this case study.\n(3) MinorillustrationchangeshavebeenmadetothefiguresintheSI.Specificallywehaveincreasedsome\nfont sizes and centered titles & labels in these plots for readability.\n(4) InSupplementaryFigure4.2,beyondtheadditionsmadeinlinewithReviewer#3’ssuggestions(added\nPanel (c), which illustrates the temporal profile of the definition and computationally efficient approx-\nimation for the objective CIR), we have also significantly increased the ε resolution of the subjective\nCIR plot (Panel (b) in the revised figure) for clarity and for better illustration.\n(5) Slightlyadjustedsomecosmeticdetailsandaddedclarificationsintheschematicdiagramfigure(Figure\n2.1 in the main text), specifically Panels (a) and (c), to help readers obtain a more intuitive idea of\nboth ACI and its conditional generalization.\n(6) We have corrected minor typos and rephrased certain sentences for clarity and readability throughout\nthe manuscript (main text and SI). All of these changes are clearly noted in the attached diff files.\n(7) We have also added brief additional clarifications and comments to the parts that needed it for expo-\nsition’s sake (main text and SI). All of these additions are clearly noted in the attached diff files.\n(8) Finally, we have also made very minor notational changes to some equations for readability (e.g.,\nadjusted parentheses sizes and replaced incorrect use of τ with τ for the subjective and objective CIR\nlengthsatsomeinstancesinthemaintextandSI).Allofthesechangesareclearlynotedintheattached\ndiff files and are purely cosmetic in nature.\nResponse to Reviewers\nRef.: Response to Referees\nManuscript Number: NCOMMS-25-48940A\nManuscript Title: Assimilative Causal Inference\nAuthors: Marios Andreou; Nan Chen; Erik Bollt\nCorresponding Author: Nan Chen\nDear Reviewers:\nWe deeply appreciate the reviewers for taking the time to thoroughly review our work and provide such\nthoughtful and insightful feedback. The reviewers’ detailed and discerning eye helped us clarify several\nvaluable points and implement additions and revisions that will further aid readers in understanding and\nappreciatingourframework. Wesincerelythankthereviewersfortheirtime,rigor,insights,andendorsement\nthrough the peer review process.\nTherevisedmanuscriptandsupplementaryinformationareattachedforyourconsideration. Inthefollowing\npages, we provide a detailed, point-by-point response to the reviewers’ comments, outlining how each has\nbeen thoughtfully considered in the revised manuscript.\nPoint-by-Point Response to Reviewers’ Comments:\nReviewer #1\nRemarks to the Authors:\nI would like to thank the authors for their detailed response and changes. I believe that the new version of\nthe paper is substantially better.\n▶ Authors’ Response: We thank the reviewer very much for their positive feedback and for all their con-\nstructivesuggestions,whichhavesignificantlyimprovedtheclarity,rigor,andqualityofourmanuscript. We\nsincerely thank them for taking the time to review our manuscript.\nHowever, some of my concerns remain:\nRegardingclaimsofscalability: Iappreciatetheauthors’clarificationthatACIisdesignedtoleverageexisting\nBayesian data assimilation machinery and hence their method can be applied in high-dimensional settings.\nHowever, such clarification is not well supported. The rebuttal does not provide a concrete complexity\nanalysis of the composite ACI method in terms of state dimension, nor does it include new experiments on\nhigher-dimensional systems. The current experiments still only contain low-dimensional examples (up to six\nvariables), which does not convincingly substantiate the strong claim that the method“scales efficiently to\nhigh-dimensional problems.”\nIwouldthereforeencouragetheauthors,atminimum,tosoftenthewordingofthescalabilityclaimthrough-\nout the paper. Perhaps the authors can emphasize that ACI can in principle be implemented in high-\ndimensional settings by reusing efficient DA algorithms, while leaving a full empirical demonstration and\nmore detailed complexity analysis to future work.\n▶ Authors’ Response: Thank you very much for the detailed comments. We have followed the reviewer’s\nsuggestion to soften the wording regarding the scalability of the method to high dimensions. Specifically, we\nmodified the abstract by adopting what the reviewer suggested:\n“It uniquely identifies dynamic causal interactions without requiring observations of candidate causes, ac-\ncommodates short datasets, and, in principle, can be implemented in high-dimensional settings by employing\nefficient data assimilation algorithms.”\nWe have also adjusted the relevant sentence in the Discussion section following the same principle. Later\nin the conclusion, we include a paragraph discussing a more detailed implementation of how Bayesian data\nassimilationcanhelpscaleACItohigh-dimensionalproblems. Thishasbeenassignedaspartoffuturework.\nThus, the tone concerning the method’s scalability has been softened over the entire manuscript.\n“By interpolating causality from effects to causes instead of extrapolating the latter forward-in-time, ACI\noffers an inherently more robust framework compared to classical predictive causality.”\nWhileIunderstandthekeydifferenceinapproach,Idonotseehowitisapparentthat“interpolatingcausality\nfrom effects to causes”results in an“inherently more robust framework”. Without sufficient theoretical or\nempirical justification, especially reading up to the point of the paper where the claim is made, it is hard to\nvalidate it.\n▶ Authors’ Response: Thank you very much for pointing this out. We did not express ourselves clearly on\nthis point. In the revised main text, we have added the following sentence after the quoted excerpt as to\nprovidefurtherexplanation. Wehaveusedtheword“typically”,hopingtoprovidegeneralempiricalevidence\nrather than a rigorous claim:\n“Interpolation is typically more stable because it operates within the range of observed information, whereas\nextrapolation requires projecting beyond available data and is therefore far more sensitive to model error and\nuncertainty.”\nRegarding real-world experiments: I appreciate the authors’ response highlighting the difficulty of working\nwith issues like addressing model error and sampling biases. I also believe that the new experiments with\nthe real-world ENSO data have made the paper stronger.\nHowever, I would still encourage the authors to add a discussion to compare the experimental results of\ntheir method to the baselines on the real-world ENSO dataset to highlight scenarios where their method\nwould perform more favorably compared to the baselines. While the theoretical contributions of the paper\nareindeedquitesignificant,practicallydemonstratingtheabilityofACItodiscoverinstantaneousedgesand\nthe resulting improvements for causal discovery performance, would further strengthen the contribution of\nthe paper.\n▶Authors’Response:Thankyouverymuchforthesuggestion. Wehaveaddedthefollowingnewparagraph\nto Section 4.3.3 of the supplementary information (“ACI on Real-World ENSO Data”):\n“Finally, it is worth emphasizing the distinction between ACI and many baseline causal inference methods\nfor analyzing real-world ENSO datasets [27, 28, 29]. Most of the current approaches are purely data-driven\nand focus on identifying broad, time-averaged cause-and-effect relationships. Their findings are generally\nconsistent with physical mechanisms, such as the causal influence of the thermocline depth on the SST and\nthedischarge–rechargelooplinkingElNin˜oandLaNin˜a. Yet, ACIprovidesamorerefinedandevent-specific\nindicator that reveals how individual variables contribute to the causal structure of each ENSO episode, an\naspectoften missingfromstate-of-the-art techniques. ACIyieldsmarkedlydifferent causalstrengthsand CIR\npatterns across events, providing valuable insight into instantaneous, time-localized cause-and-effect behav-\nior. As a natural direction for future work, the ACI framework can be coupled with a range of ENSO models\nand expanded to include additional variables, thereby enhancing causal discovery with a particular focus on\ninstantaneous relationships.”\nReviewer #2\nRemarks to the Authors:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the\nNatureCommunicationsinitiativetofacilitatetraininginpeerreviewandtoprovideappropriaterecognition\nfor Early Career Researchers who co-review manuscripts.\n▶ Authors’ Response: We sincerely thank the reviewer again for their efforts and constructive feedback pro-\nvided under the Nature Communications co-review initiative for early career researchers. Their input and\ntime is greatly appreciated.\nReviewer #3\nRemarks to the Authors:\nDear editor and authors,\nThis is my response to the revised manuscript entitled ”Assimilative Causal Inference.”\nI want to thank the authors for a very thorough, detailed and easy-to-read response letter and for providing\ndiff-files making it clear what has been changed.\nAfter reviewing the revised manuscript, I am satisfied that my initial comments have all been adequately\naddressed by the authors. Specifically, I am happy to see that the authors have provided a repository with\nthe code used to produce the results in the manuscript.\nIrecommendpublicationofthisrevisedmanuscript, andIlookforwardtomoreresearchonthemethodand\nto see applications by other researchers.\nRegards,\nDan Mønster\n▶ Authors’ Response: We are grateful to the reviewer once again for their insightful and constructive com-\nments, which have greatly enhanced the clarity, rigor, and quality of our manuscript. We sincerely thank\nthem for taking the time to review our manuscript.\nReviewer #4\nRemarks to the Authors:\nIthanktheauthorsfortheirthoroughresponsetomyreportandthoseoftheotherrefereesandtheassociated\nrevisions of the text, and am pleased to recommend the paper for publication.\n▶ Authors’ Response: We thank the reviewer once again for their highly constructive feedback and sugges-\ntions,whichhavesignificantlyimprovedtheclarity,rigor,andqualityofourmanuscript. Wesincerelythank\nthem for taking the time to review our manuscript.\nWe sincerely thank all of the reviewers for their time and suggestions!"}]

Decision letter