The challenge of identifying causes and effects in complex systems, such as climate or the brain, has long been hampered by a limitation: most methods required either intervening in the system or representing relationships as simple graphs. However, many natural systems do not allow interventions, and their interactions are too rich for graphs where links are only pairwise. Alongside this work on the archive are studies of quantum map detectives and invisible links in gene networks, but they concern different objects. This paper fills the gap by proposing a unified framework for non-interventional systems with polyadic interactions, where multiple causes act jointly.
The authors build the framework on information theory, but instead of entropy they introduce "certainty," a non-negative measure of how much we know about a process relative to a broad reference distribution. The total mutual information between the target and all drivers is decomposed into direct contributions of each driver (1link) and joint contributions of pairs, triples, and so on, subtracting lower-order contributions to avoid double counting. Contributions are normalized by conditional certainty, giving causal strength in percentages and allowing comparisons across different studies. To estimate information measures, the k-nearest neighbors algorithm is used, and the reference distribution is chosen as a Cauchy distribution with a width matched to the entropy of the Gaussian distribution of the data.
On simple linear and nonlinear models, the framework correctly identified direct and mediated links, as well as contributions from unknown processes. For example, in a gate model (y → z → x), the method showed that y influences x only through z, while in a common driver model (y ← z → x), the causal strength of y was zero. On the chaotic Lorenz 1963 system, the method found that joint contributions (2links) are much stronger than direct ones: knowing x and y together gives 9 times more information about the future of x than knowing x alone. Application to El Niño data (Niño3.4, U850, U200, HEAT indices for 1979–2019) showed that the largest contribution to sea surface temperature comes from the heat content of the upper 300 meters (25% causal strength), and joint interactions amplify the contribution of each factor roughly twofold compared to direct links.
Unlike neighboring works, which are either limited to directed acyclic graphs or do not provide a full decomposition, this framework for the first time allows decomposing causal influence into all orders of interactions and estimating the contribution of unknown processes without additional data. This is a step from binary links to polyadic ones, which is important for atmospheric physics, oceanography, and other systems with strong feedbacks. The method also shows that ignoring joint contributions can change the ranking of driver importance.
The framework can be extended to non-stationary time series, for example to study climate change or financial regimes. The authors plan to add Bayesian uncertainty estimation to test hypotheses of zero causal strength. Also promising is the combination with symbolic transfer entropy methods for non-stationary data.
The method is applicable in Earth sciences (climate, ocean), astrophysics, neuroscience, and other fields where only observational time series are available and interventions are impossible.
Immediate next steps include developing a Bayesian version to estimate uncertainties and adapting to non-stationary series, as well as deeper investigation of the physical meaning of joint contributions on real data.
In statistics and machine learning, an open question remains: how to define causality in systems with a continuous set of features and without interventions. This work is directly related to the problem of identifiability of causal structures from observational data and to the limitations of graphical models.
On the Lorenz 1963 model, the method showed that knowing coordinates x and y together gives 9 times more information about the future of x than knowing x alone, even though y does not appear in the equation for x.