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Nathan Mankovich, Homer Durand, Emiliano Diaz, Gherardo Varando, Gustau Camps-Valls

Detecting latent confounders from proxy variables is an essential problem in causal effect estimation. Previous approaches are limited to low-dimensional proxies, sorted proxies, and binary treatments. We remove these assumptions and present a novel Proxy Confounder Factorization (PCF) framework for continuous treatment effect estimation when latent confounders manifest through high-dimensional, mixed proxy variables. For specific sample sizes, our two-step PCF implementation, using Independent Component Analysis (ICA-PCF), and the end-to-end implementation, using Gradient Descent (GD-PCF), achieve high correlation with the latent confounder and low absolute error in causal effect estimation with synthetic datasets in the high sample size regime. Even when faced with climate data, ICA-PCF recovers four components that explain $75.9\%$ of the variance in the North Atlantic Oscillation, a known confounder of precipitation patterns in Europe. Code for our PCF implementations and experiments can be found here: https://github.com/IPL-UV/confound_it. The proposed methodology constitutes a stepping stone towards discovering latent confounders and can be applied to many problems in disciplines dealing with high-dimensional observed proxies, e.g., spatiotemporal fields.

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Homer Durand, Gherardo Varando, Nathan Mankovich, Gustau Camps-Valls

We introduce a causal regularisation extension to anchor regression (AR) for improved out-of-distribution (OOD) generalisation. We present anchor-compatible losses, aligning with the anchor framework to ensure robustness against distribution shifts. Various multivariate analysis (MVA) algorithms, such as (Orthonormalized) PLS, RRR, and MLR, fall within the anchor framework. We observe that simple regularisation enhances robustness in OOD settings. Estimators for selected algorithms are provided, showcasing consistency and efficacy in synthetic and real-world climate science problems. The empirical validation highlights the versatility of anchor regularisation, emphasizing its compatibility with MVA approaches and its role in enhancing replicability while guarding against distribution shifts. The extended AR framework advances causal inference methodologies, addressing the need for reliable OOD generalisation.

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Kai-Hendrik Cohrs, Gherardo Varando, Nuno Carvalhais, Markus Reichstein, Gustau Camps-Valls

Hybrid modeling integrates machine learning with scientific knowledge with the goal of enhancing interpretability, generalization, and adherence to natural laws. Nevertheless, equifinality and regularization biases pose challenges in hybrid modeling to achieve these purposes. This paper introduces a novel approach to estimating hybrid models via a causal inference framework, specifically employing Double Machine Learning (DML) to estimate causal effects. We showcase its use for the Earth sciences on two problems related to carbon dioxide fluxes. In the $Q_{10}$ model, we demonstrate that DML-based hybrid modeling is superior in estimating causal parameters over end-to-end deep neural network (DNN) approaches, proving efficiency, robustness to bias from regularization methods, and circumventing equifinality. Our approach, applied to carbon flux partitioning, exhibits flexibility in accommodating heterogeneous causal effects. The study emphasizes the necessity of explicitly defining causal graphs and relationships, advocating for this as a general best practice. We encourage the continued exploration of causality in hybrid models for more interpretable and trustworthy results in knowledge-guided machine learning.

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Gustau Camps-Valls, Andreas Gerhardus, Urmi Ninad, Gherardo Varando, Georg Martius, Emili Balaguer-Ballester, Ricardo Vinuesa, Emiliano Diaz, Laure Zanna, Jakob Runge

Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws and principles that are invariant, robust and causal explanations of the world has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventional studies in the system under study. With the advent of big data and the use of data-driven methods, causal and equation discovery fields have grown and made progress in computer science, physics, statistics, philosophy, and many applied fields. All these domains are intertwined and can be used to discover causal relations, physical laws, and equations from observational data. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of Physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for observational causal and equation discovery, point out connections, and showcase a complete set of case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is being revolutionised with the efficient exploitation of observational data, modern machine learning algorithms and the interaction with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems.

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Manuele Leonelli, Gherardo Varando

Bayesian networks are widely used to learn and reason about the dependence structure of discrete variables. However, they are only capable of formally encoding symmetric conditional independence, which in practice is often too strict to hold. Asymmetry-labeled DAGs have been recently proposed to both extend the class of Bayesian networks by relaxing the symmetric assumption of independence and denote the type of dependence existing between the variables of interest. Here, we introduce novel structural learning algorithms for this class of models which, whilst being efficient, allow for a straightforward interpretation of the underlying dependence structure. A comprehensive computational study highlights the efficiency of the algorithms. A real-world data application using data from the Fear of COVID-19 Scale collected in Italy showcases their use in practice.

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Manuele Leonelli, Gherardo Varando

Several structural learning algorithms for staged tree models, an asymmetric extension of Bayesian networks, have been defined. However, they do not scale efficiently as the number of variables considered increases. Here we introduce the first scalable structural learning algorithm for staged trees, which searches over a space of models where only a small number of dependencies can be imposed. A simulation study as well as a real-world application illustrate our routines and the practical use of such data-learned staged trees.

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Manuele Leonelli, Gherardo Varando

Bayesian networks faithfully represent the symmetric conditional independences existing between the components of a random vector. Staged trees are an extension of Bayesian networks for categorical random vectors whose graph represents non-symmetric conditional independences via vertex coloring. However, since they are based on a tree representation of the sample space, the underlying graph becomes cluttered and difficult to visualize as the number of variables increases. Here we introduce the first structural learning algorithms for the class of simple staged trees, entertaining a compact coalescence of the underlying tree from which non-symmetric independences can be easily read. We show that data-learned simple staged trees often outperform Bayesian networks in model fit and illustrate how the coalesced graph is used to identify non-symmetric conditional independences.

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Gherardo Varando, Federico Carli, Manuele Leonelli

Bayesian networks are a widely-used class of probabilistic graphical models capable of representing symmetric conditional independence between variables of interest using the topology of the underlying graph. They can be seen as a special case of the much more general class of models called staged trees, which can represent any type of non-symmetric conditional independence. Here we formalize the relationship between these two models and introduce a minimal Bayesian network representation of the staged tree, which can be used to read conditional independences in an intuitive way. Furthermore, we define a new labeled graph, termed asymmetry-labeled directed acyclic graph, whose edges are labeled to denote the type of dependence existing between any two random variables. Various datasets are used to illustrate the methodology, highlighting the need to construct models which more flexibly encode and represent non-symmetric structures.

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Manuele Leonelli, Gherardo Varando

Causal discovery algorithms aims at untangling complex causal relationships using observational data only. Here, we introduce new causal discovery algorithms based on staged tree models, which can represent complex and non-symmetric causal effects. To demonstrate the efficacy of our algorithms, we introduce a new distance, inspired by the widely used structural interventional distance, to quantify the closeness between two staged trees in terms of their corresponding causal inference statements. A simulation study highlights the efficacy of staged trees in uncovering complex, asymmetric causal relationship from data and a real-world data application illustrates their use in a practical causal analysis.

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Federico Carli, Manuele Leonelli, Gherardo Varando

Generative models for classification use the joint probability distribution of the class variable and the features to construct a decision rule. Among generative models, Bayesian networks and naive Bayes classifiers are the most commonly used and provide a clear graphical representation of the relationship among all variables. However, these have the disadvantage of highly restricting the type of relationships that could exist, by not allowing for context-specific independences. Here we introduce a new class of generative classifiers, called staged tree classifiers, which formally account for context-specific independence. They are constructed by a partitioning of the vertices of an event tree from which conditional independence can be formally read. The naive staged tree classifier is also defined, which extends the classic naive Bayes classifier whilst retaining the same complexity. An extensive simulation study shows that the classification accuracy of staged tree classifiers is competitive with those of state-of-the-art classifiers. An applied analysis to predict the fate of the passengers of the Titanic highlights the insights that the new class of generative classifiers can give.

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