Peter Bühlmann

Alexander Immer, Christoph Schultheiss, Julia E. Vogt et al.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect $Y$ can be written as a function of the cause $X$ and a noise source $N$ independent of $X$, which may be scaled by a positive function $g$ over the cause, i.e., $Y = f(X) + g(X)N$. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of $Y$ given $X$ as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

Xinwei Shen, Peter Bühlmann, Armeen Taeb

Since distribution shifts are common in real-world applications, there is a pressing need to develop prediction models that are robust against such shifts. Existing frameworks, such as empirical risk minimization or distributionally robust optimization, either lack generalizability for unseen distributions or rely on postulated distance measures. Alternatively, causality offers a data-driven and structural perspective to robust predictions. However, the assumptions necessary for causal inference can be overly stringent, and the robustness offered by such causal models often lacks flexibility. In this paper, we focus on causality-oriented robustness and propose Distributional Robustness via Invariant Gradients (DRIG), a method that exploits general noise interventions in training data for robust predictions against unseen interventions, and naturally interpolates between in-distribution prediction and causality. In a linear setting, we prove that DRIG yields predictions that are robust among a data-dependent class of distribution shifts. Furthermore, we show that our framework includes anchor regression as a special case, and that it yields prediction models that protect against more diverse perturbations. We establish finite-sample results and extend our approach to semi-supervised domain adaptation to further improve prediction performance. Finally, we empirically validate our methods on synthetic simulations and on single-cell and intensive health care datasets.

Michael J. Zellinger, Peter Bühlmann

Cluster analysis relies on effective benchmarks for evaluating and comparing different algorithms. Simulation studies on synthetic data are popular because important features of the data sets, such as the overlap between clusters, or the variation in cluster shapes, can be effectively varied. Unfortunately, creating evaluation scenarios is often laborious, as practitioners must translate higher-level scenario descriptions like "clusters with very different shapes" into lower-level geometric parameters such as cluster centers, covariance matrices, etc. To make benchmarks more convenient and informative, we propose synthetic data generation based on direct specification of high-level scenarios, either through verbal descriptions or high-level geometric parameters. Our open-source Python package repliclust implements this workflow, making it easy to set up interpretable and reproducible benchmarks for cluster analysis. A demo of data generation from verbal inputs is available at https://demo.repliclust.org.

Zhenyu Wang, Peter Bühlmann, Zijian Guo

Empirical risk minimization often performs poorly when the distribution of the target domain differs from those of source domains. To address such potential distribution shifts, we develop an unsupervised domain adaptation approach that leverages labeled data from multiple source domains and unlabeled data from the target domain. We introduce a distributionally robust model that optimizes an adversarial reward based on the explained variance across a class of target distributions, ensuring generalization to the target domain. We show that the proposed robust model is a weighted average of conditional outcome models from source domains. This formulation allows us to compute the robust model through the aggregation of source models, which can be estimated using various machine learning algorithms of the users' choice, such as random forests, boosting, and neural networks. Additionally, we introduce a bias-correction step to obtain a more accurate aggregation weight, which is effective for various machine learning algorithms. Our framework can be interpreted as a distributionally robust federated learning approach that satisfies privacy constraints while providing insights into the importance of each source for prediction on the target domain. The performance of our method is evaluated on both simulated and real data.

Juan L. Gamella, Jonas Peters, Peter Bühlmann · eth-zurich

In some fields of AI, machine learning and statistics, the validation of new methods and algorithms is often hindered by the scarcity of suitable real-world datasets. Researchers must often turn to simulated data, which yields limited information about the applicability of the proposed methods to real problems. As a step forward, we have constructed two devices that allow us to quickly and inexpensively produce large datasets from non-trivial but well-understood physical systems. The devices, which we call causal chambers, are computer-controlled laboratories that allow us to manipulate and measure an array of variables from these physical systems, providing a rich testbed for algorithms from a variety of fields. We illustrate potential applications through a series of case studies in fields such as causal discovery, out-of-distribution generalization, change point detection, independent component analysis, and symbolic regression. For applications to causal inference, the chambers allow us to carefully perform interventions. We also provide and empirically validate a causal model of each chamber, which can be used as ground truth for different tasks. All hardware and software is made open source, and the datasets are publicly available at causalchamber.org or through the Python package causalchamber.

Alexander Henzi, Xinwei Shen, Michael Law et al.

In recent years, there has been a growing interest in statistical methods that exhibit robust performance under distribution changes between training and test data. While most of the related research focuses on point predictions with the squared error loss, this article turns the focus towards probabilistic predictions, which aim to comprehensively quantify the uncertainty of an outcome variable given covariates. Within a causality-inspired framework, we investigate the invariance and robustness of probabilistic predictions with respect to proper scoring rules. We show that arbitrary distribution shifts do not, in general, admit invariant and robust probabilistic predictions, in contrast to the setting of point prediction. We illustrate how to choose evaluation metrics and restrict the class of distribution shifts to allow for identifiability and invariance in the prototypical Gaussian heteroscedastic linear model. Motivated by these findings, we propose a method to yield invariant probabilistic predictions, called IPP, and study the consistency of the underlying parameters. Finally, we demonstrate the empirical performance of our proposed procedure on simulated as well as on single-cell data.

Domagoj Ćevid, Loris Michel, Jeffrey Näf et al.

Random Forest (Breiman, 2001) is a successful and widely used regression and classification algorithm. Part of its appeal and reason for its versatility is its (implicit) construction of a kernel-type weighting function on training data, which can also be used for targets other than the original mean estimation. We propose a novel forest construction for multivariate responses based on their joint conditional distribution, independent of the estimation target and the data model. It uses a new splitting criterion based on the MMD distributional metric, which is suitable for detecting heterogeneity in multivariate distributions. The induced weights define an estimate of the full conditional distribution, which in turn can be used for arbitrary and potentially complicated targets of interest. The method is very versatile and convenient to use, as we illustrate on a wide range of examples. The code is available as Python and R packages drf.

Jonas Peters, Peter Bühlmann

Causal inference relies on the structure of a graph, often a directed acyclic graph (DAG). Different graphs may result in different causal inference statements and different intervention distributions. To quantify such differences, we propose a (pre-) distance between DAGs, the structural intervention distance (SID). The SID is based on a graphical criterion only and quantifies the closeness between two DAGs in terms of their corresponding causal inference statements. It is therefore well-suited for evaluating graphs that are used for computing interventions. Instead of DAGs it is also possible to compare CPDAGs, completed partially directed acyclic graphs that represent Markov equivalence classes. Since it differs significantly from the popular Structural Hamming Distance (SHD), the SID constitutes a valuable additional measure. We discuss properties of this distance and provide an efficient implementation with software code available on the first author's homepage (an R package is under construction).

Yihong Gu, Cong Fang, Peter Bühlmann et al. · princeton

Pursuing causality from data is a fundamental problem in scientific discovery, treatment intervention, and transfer learning. This paper introduces a novel algorithmic method for addressing nonparametric invariance and causality learning in regression models across multiple environments, where the joint distribution of response variables and covariates varies, but the conditional expectations of outcome given an unknown set of quasi-causal variables are invariant. The challenge of finding such an unknown set of quasi-causal or invariant variables is compounded by the presence of endogenous variables that have heterogeneous effects across different environments. The proposed Focused Adversarial Invariant Regularization (FAIR) framework utilizes an innovative minimax optimization approach that drives regression models toward prediction-invariant solutions through adversarial testing. Leveraging the representation power of neural networks, FAIR neural networks (FAIR-NN) are introduced for causality pursuit. It is shown that FAIR-NN can find the invariant variables and quasi-causal variables under a minimal identification condition and that the resulting procedure is adaptive to low-dimensional composition structures in a non-asymptotic analysis. Under a structural causal model, variables identified by FAIR-NN represent pragmatic causality and provably align with exact causal mechanisms under conditions of sufficient heterogeneity. Computationally, FAIR-NN employs a novel Gumbel approximation with decreased temperature and a stochastic gradient descent ascent algorithm. The procedures are demonstrated using simulated and real-data examples.

Zhenyu Wang, Yifan Hu, Peter Bühlmann et al.

Data from multiple environments offer valuable opportunities to uncover causal relationships among variables. Leveraging the assumption that the causal outcome model remains invariant across heterogeneous environments, state-of-the-art methods attempt to identify causal outcome models by learning invariant prediction models and rely on exhaustive searches over all (exponentially many) covariate subsets. These approaches present two major challenges: 1) determining the conditions under which the invariant prediction model aligns with the causal outcome model, and 2) devising computationally efficient causal discovery algorithms that scale polynomially, instead of exponentially, with the number of covariates. To address both challenges, we focus on the additive intervention regime and propose nearly necessary and sufficient conditions for ensuring that the invariant prediction model matches the causal outcome model. Exploiting the essentially necessary identifiability conditions, we introduce Negative Weight Distributionally Robust Optimization (NegDRO), a nonconvex continuous minimax optimization whose global optimizer recovers the causal outcome model. Unlike standard group DRO problems that maximize over the simplex, NegDRO allows negative weights on environment losses, which break the convexity. Despite its nonconvexity, we demonstrate that a standard gradient method converges to the causal outcome model, and we establish the convergence rate with respect to the sample size and the number of iterations. Our algorithm avoids exhaustive search, making it scalable especially when the number of covariates is large. The numerical results further validate the efficiency of the proposed method.

Marin Šola, Peter Bühlmann, Xinwei Shen

Distributional robustness is a central goal of prediction algorithms due to the prevalent distribution shifts in real-world data. The prediction model aims to minimize the worst-case risk among a class of distributions, a.k.a., an uncertainty set. Causality provides a modeling framework with a rigorous robustness guarantee in the above sense, where the uncertainty set is data-driven rather than pre-specified as in traditional distributional robustness optimization. However, current causality-inspired robustness methods possess finite-radius robustness guarantees only in the linear settings, where the causal relationships among the covariates and the response are linear. In this work, we propose a nonlinear method under a causal framework by incorporating recent developments in identifiable representation learning and establish a distributional robustness guarantee. To our best knowledge, this is the first causality-inspired robustness method with such a finite-radius robustness guarantee in nonlinear settings. Empirical validation of the theoretical findings is conducted on both synthetic data and real-world single-cell data, also illustrating that finite-radius robustness is crucial.

Malte Londschien, Manuel Burger, Gunnar Rätsch et al.

The performance of predictive models in clinical settings often degrades when deployed in new hospitals due to distribution shifts. This paper presents a large-scale study of causality-inspired domain generalization on heterogeneous multi-center intensive care unit (ICU) data. We apply anchor regression and introduce anchor boosting, a novel, tree-based nonlinear extension, to a large dataset comprising 400,000 patients from nine distinct ICU databases. The anchor regularization consistently improves out-of-distribution performance, particularly for the most dissimilar target domains. The methods appear robust to violations of theoretical assumptions, such as anchor exogeneity. Furthermore, we propose a novel conceptual framework to quantify the utility of large external data datasets. By evaluating performance as a function of available target-domain data, we identify three regimes: (i) a domain generalization regime, where only the external model should be used, (ii) a domain adaptation regime, where refitting the external model is optimal, and (iii) a data-rich regime, where external data provides no additional value.

Mona Azadkia, Armeen Taeb, Peter Bühlmann

We study the problem of causal structure learning with essentially no assumptions on the functional relationships and noise. We develop DAG-FOCI, a computationally fast algorithm for this setting that is based on the FOCI variable selection algorithm in~\cite{azadkia2021simple}. DAG-FOCI outputs the set of parents of a response variable of interest. We provide theoretical guarantees of our procedure when the underlying graph does not contain any (undirected) cycle containing the response variable of interest. Furthermore, in the absence of this assumption, we give a conservative guarantee against false positive causal claims when the set of parents is identifiable. We demonstrate the applicability of DAG-FOCI on simulated as well as a real dataset from computational biology~\cite{sachs2005causal}.

Corinne Emmenegger, Peter Bühlmann

Traditionally, spline or kernel approaches in combination with parametric estimation are used to infer the linear coefficient (fixed effects) in a partially linear mixed-effects model for repeated measurements. Using machine learning algorithms allows us to incorporate complex interaction structures and high-dimensional variables. We employ double machine learning to cope with the nonparametric part of the partially linear mixed-effects model: the nonlinear variables are regressed out nonparametrically from both the linear variables and the response. This adjustment can be performed with any machine learning algorithm, for instance random forests, which allows to take complex interaction terms and nonsmooth structures into account. The adjusted variables satisfy a linear mixed-effects model, where the linear coefficient can be estimated with standard linear mixed-effects techniques. We prove that the estimated fixed effects coefficient converges at the parametric rate, is asymptotically Gaussian distributed, and semiparametrically efficient. Two simulation studies demonstrate that our method outperforms a penalized regression spline approach in terms of coverage. We also illustrate our proposed approach on a longitudinal dataset with HIV-infected individuals. Software code for our method is available in the R-package dmlalg.

Martin Emil Jakobsen, Rajen D. Shah, Peter Bühlmann et al.

Knowing the causal structure of a system is of fundamental interest in many areas of science and can aid the design of prediction algorithms that work well under manipulations to the system. The causal structure becomes identifiable from the observational distribution under certain restrictions. To learn the structure from data, score-based methods evaluate different graphs according to the quality of their fits. However, for large, continuous, and nonlinear models, these rely on heuristic optimization approaches with no general guarantees of recovering the true causal structure. In this paper, we consider structure learning of directed trees. We propose a fast and scalable method based on Chu-Liu-Edmonds' algorithm we call causal additive trees (CAT). For the case of Gaussian errors, we prove consistency in an asymptotic regime with a vanishing identifiability gap. We also introduce two methods for testing substructure hypotheses with asymptotic family-wise error rate control that is valid post-selection and in unidentified settings. Furthermore, we study the identifiability gap, which quantifies how much better the true causal model fits the observational distribution, and prove that it is lower bounded by local properties of the causal model. Simulation studies demonstrate the favorable performance of CAT compared to competing structure learning methods.

Michael Moor, Nicolas Bennet, Drago Plecko et al.

Despite decades of clinical research, sepsis remains a global public health crisis with high mortality, and morbidity. Currently, when sepsis is detected and the underlying pathogen is identified, organ damage may have already progressed to irreversible stages. Effective sepsis management is therefore highly time-sensitive. By systematically analysing trends in the plethora of clinical data available in the intensive care unit (ICU), an early prediction of sepsis could lead to earlier pathogen identification, resistance testing, and effective antibiotic and supportive treatment, and thereby become a life-saving measure. Here, we developed and validated a machine learning (ML) system for the prediction of sepsis in the ICU. Our analysis represents the largest multi-national, multi-centre in-ICU study for sepsis prediction using ML to date. Our dataset contains $156,309$ unique ICU admissions, which represent a refined and harmonised subset of five large ICU databases originating from three countries. Using the international consensus definition Sepsis-3, we derived hourly-resolved sepsis label annotations, amounting to $26,734$ ($17.1\%$) septic stays. We compared our approach, a deep self-attention model, to several clinical baselines as well as ML baselines and performed an extensive internal and external validation within and across databases. On average, our model was able to predict sepsis with an AUROC of $0.847 \pm 0.050$ (internal out-of sample validation) and $0.761 \pm 0.052$ (external validation). For a harmonised prevalence of $17\%$, at $80\%$ recall our model detects septic patients with $39\%$ precision 3.7 hours in advance.

Yuansi Chen, Peter Bühlmann

Domain adaptation (DA) arises as an important problem in statistical machine learning when the source data used to train a model is different from the target data used to test the model. Recent advances in DA have mainly been application-driven and have largely relied on the idea of a common subspace for source and target data. To understand the empirical successes and failures of DA methods, we propose a theoretical framework via structural causal models that enables analysis and comparison of the prediction performance of DA methods. This framework also allows us to itemize the assumptions needed for the DA methods to have a low target error. Additionally, with insights from our theory, we propose a new DA method called CIRM that outperforms existing DA methods when both the covariates and label distributions are perturbed in the target data. We complement the theoretical analysis with extensive simulations to show the necessity of the devised assumptions. Reproducible synthetic and real data experiments are also provided to illustrate the strengths and weaknesses of DA methods when parts of the assumptions in our theory are violated.

Solt Kovács, Housen Li, Lorenz Haubner et al.

Change point estimation is often formulated as a search for the maximum of a gain function describing improved fits when segmenting the data. Searching through all candidates requires $O(n)$ evaluations of the gain function for an interval with $n$ observations. If each evaluation is computationally demanding (e.g. in high-dimensional models), this can become infeasible. Instead, we propose optimistic search methods with $O(\log n)$ evaluations exploiting specific structure of the gain function. Towards solid understanding of our strategy, we investigate in detail the $p$-dimensional Gaussian changing means setup, including high-dimensional scenarios. For some of our proposals, we prove asymptotic minimax optimality for detecting change points and derive their asymptotic localization rate. These rates (up to a possible log factor) are optimal for the univariate and multivariate scenarios, and are by far the fastest in the literature under the weakest possible detection condition on the signal-to-noise ratio in the high-dimensional scenario. Computationally, our proposed methodology has the worst case complexity of $O(np)$, which can be improved to be sublinear in $n$ if some a-priori knowledge on the length of the shortest segment is available. Our search strategies generalize far beyond the theoretically analyzed setup. We illustrate, as an example, massive computational speedup in change point detection for high-dimensional Gaussian graphical models.

Peter Bühlmann, Domagoj Ćevid

We review some recent work on removing hidden confounding and causal regularization from a unified viewpoint. We describe how simple and user-friendly techniques improve stability, replicability and distributional robustness in heterogeneous data. In this sense, we provide additional thoughts to the issue on concept drift, raised by Efron (2020), when the data generating distribution is changing.

Malte Londschien, Solt Kovács, Peter Bühlmann

We propose estimation methods for change points in high-dimensional covariance structures with an emphasis on challenging scenarios with missing values. We advocate three imputation like methods and investigate their implications on common losses used for change point detection. We also discuss how model selection methods have to be adapted to the setting of incomplete data. The methods are compared in a simulation study and applied to a time series from an environmental monitoring system. An implementation of our proposals within the R-package hdcd is available via the Supplementary materials.

Niklas Pfister, Sebastian Weichwald, Peter Bühlmann et al.

We introduce coroICA, confounding-robust independent component analysis, a novel ICA algorithm which decomposes linearly mixed multivariate observations into independent components that are corrupted (and rendered dependent) by hidden group-wise stationary confounding. It extends the ordinary ICA model in a theoretically sound and explicit way to incorporate group-wise (or environment-wise) confounding. We show that our proposed general noise model allows to perform ICA in settings where other noisy ICA procedures fail. Additionally, it can be used for applications with grouped data by adjusting for different stationary noise within each group. Our proposed noise model has a natural relation to causality and we explain how it can be applied in the context of causal inference. In addition to our theoretical framework, we provide an efficient estimation procedure and prove identifiability of the unmixing matrix under mild assumptions. Finally, we illustrate the performance and robustness of our method on simulated data, provide audible and visual examples, and demonstrate the applicability to real-world scenarios by experiments on publicly available Antarctic ice core data as well as two EEG data sets. We provide a scikit-learn compatible pip-installable Python package coroICA as well as R and Matlab implementations accompanied by a documentation at https://sweichwald.de/coroICA/

Niklas Pfister, Peter Bühlmann, Bernhard Schölkopf et al.

We investigate the problem of testing whether $d$ random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the $d$-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the $d$ variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small $d$, it performs well in practice. Finally, we apply the test to a problem in causal discovery.

Christopher Nowzohour, Marloes H. Maathuis, Robin J. Evans et al.

We consider the problem of structure learning for bow-free acyclic path diagrams (BAPs). BAPs can be viewed as a generalization of linear Gaussian DAG models that allow for certain hidden variables. We present a first method for this problem using a greedy score-based search algorithm. We also prove some necessary and some sufficient conditions for distributional equivalence of BAPs which are used in an algorithmic ap- proach to compute (nearly) equivalent model structures. This allows us to infer lower bounds of causal effects. We also present applications to real and simulated datasets using our publicly available R-package.

Jonas Peters, Peter Bühlmann, Nicolai Meinshausen

What is the difference of a prediction that is made with a causal model and a non-causal model? Suppose we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (for example various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid confidence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable. We further investigate robustness properties of our approach under model misspecification and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.

Christopher Nowzohour, Peter Bühlmann

Given data sampled from a number of variables, one is often interested in the underlying causal relationships in the form of a directed acyclic graph. In the general case, without interventions on some of the variables it is only possible to identify the graph up to its Markov equivalence class. However, in some situations one can find the true causal graph just from observational data, for example in structural equation models with additive noise and nonlinear edge functions. Most current methods for achieving this rely on nonparametric independence tests. One of the problems there is that the null hypothesis is independence, which is what one would like to get evidence for. We take a different approach in our work by using a penalized likelihood as a score for model selection. This is practically feasible in many settings and has the advantage of yielding a natural ranking of the candidate models. When making smoothness assumptions on the probability density space, we prove consistency of the penalized maximum likelihood estimator. We also present empirical results for simulated scenarios and real two-dimensional data sets (cause-effect pairs) where we obtain similar results as other state-of-the-art methods.

Po-Ling Loh, Peter Bühlmann

We establish a new framework for statistical estimation of directed acyclic graphs (DAGs) when data are generated from a linear, possibly non-Gaussian structural equation model. Our framework consists of two parts: (1) inferring the moralized graph from the support of the inverse covariance matrix; and (2) selecting the best-scoring graph amongst DAGs that are consistent with the moralized graph. We show that when the error variances are known or estimated to close enough precision, the true DAG is the unique minimizer of the score computed using the reweighted squared l_2-loss. Our population-level results have implications for the identifiability of linear SEMs when the error covariances are specified up to a constant multiple. On the statistical side, we establish rigorous conditions for high-dimensional consistency of our two-part algorithm, defined in terms of a "gap" between the true DAG and the next best candidate. Finally, we demonstrate that dynamic programming may be used to select the optimal DAG in linear time when the treewidth of the moralized graph is bounded.

Peter Bühlmann, Jonas Peters, Jan Ernest

We develop estimation for potentially high-dimensional additive structural equation models. A key component of our approach is to decouple order search among the variables from feature or edge selection in a directed acyclic graph encoding the causal structure. We show that the former can be done with nonregularized (restricted) maximum likelihood estimation while the latter can be efficiently addressed using sparse regression techniques. Thus, we substantially simplify the problem of structure search and estimation for an important class of causal models. We establish consistency of the (restricted) maximum likelihood estimator for low- and high-dimensional scenarios, and we also allow for misspecification of the error distribution. Furthermore, we develop an efficient computational algorithm which can deal with many variables, and the new method's accuracy and performance is illustrated on simulated and real data.

Jonas Peters, Peter Bühlmann

We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model, there is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes, assuming faithfulness. In this work, we prove full identifiability if all noise variables have the same variances: the directed acyclic graph can be recovered from the joint Gaussian distribution. Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with equal error variances and assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and an algorithm that exploit our theoretical findings.