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Christoph Schultheiss, Peter Bühlmann

We propose a method to detect model misspecifications in nonlinear causal additive and potentially heteroscedastic noise models. We aim to identify predictor variables for which we can infer the causal effect even in cases of such misspecification. We develop a general framework based on knowledge of the multivariate observational data distribution and we then propose an algorithm for finite sample data, discuss its asymptotic properties, and illustrate its performance on simulated and real data.

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Alexander Henzi, Xinwei Shen, Michael Law, Peter Bühlmann

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.

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Zhenyu Wang, Peter Bühlmann, Zijian Guo

Classical machine learning methods may lead to poor prediction performance when the target distribution differs from the source populations. This paper utilizes data from multiple sources and introduces a group distributionally robust prediction model defined to optimize an adversarial reward about explained variance with respect to a class of target distributions. Compared to classical empirical risk minimization, the proposed robust prediction model improves the prediction accuracy for target populations with distribution shifts. We show that our group distributionally robust prediction model is a weighted average of the source populations' conditional outcome models. We leverage this key identification result to robustify arbitrary machine learning algorithms, including, for example, random forests and neural networks. We devise a novel bias-corrected estimator to estimate the optimal aggregation weight for general machine-learning algorithms and demonstrate its improvement in the convergence rate. Our proposal can be seen as a distributionally robust federated learning approach that is computationally efficient and easy to implement using arbitrary machine learning base algorithms, satisfies some privacy constraints, and has a nice interpretation of different sources' importance for predicting a given target covariate distribution. We demonstrate the performance of our proposed group distributionally robust method on simulated and real data with random forests and neural networks as base-learning algorithms.

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Xinwei Shen, Peter Bühlmann, Armeen Taeb

Since distribution shifts are common in real-world applications, there is a pressing need for developing 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 additive 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 (Rothenh\"ausler et al.\ 2021) as a special case, and that it yields prediction models that protect against more diverse perturbations. We extend our approach to the semi-supervised domain adaptation setting to further improve prediction performance. Finally, we empirically validate our methods on synthetic simulations and on single-cell data.

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David Carl, Corinne Emmenegger, Peter Bühlmann, Zijian Guo

TSCI implements treatment effect estimation from observational data under invalid instruments in the R statistical computing environment. Existing instrumental variable approaches rely on arguably strong and untestable identification assumptions, which limits their practical application. TSCI does not require the classical instrumental variable identification conditions and is effective even if all instruments are invalid. TSCI implements a two-stage algorithm. In the first stage, machine learning is used to cope with nonlinearities and interactions in the treatment model. In the second stage, a space to capture the instrument violations is selected in a data-adaptive way. These violations are then projected out to estimate the treatment effect.

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Michael J. Zellinger, Peter Bühlmann

We present repliclust (from repli-cate and clust-er), a Python package for generating synthetic data sets with clusters. Our approach is based on data set archetypes, high-level geometric descriptions from which the user can create many different data sets, each possessing the desired geometric characteristics. The architecture of our software is modular and object-oriented, decomposing data generation into algorithms for placing cluster centers, sampling cluster shapes, selecting the number of data points for each cluster, and assigning probability distributions to clusters. The project webpage, repliclust.org, provides a concise user guide and thorough documentation.

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Jeffrey Näf, Corinne Emmenegger, Peter Bühlmann, Nicolai Meinshausen

The Distributional Random Forest (DRF) is a recently introduced Random Forest algorithm to estimate multivariate conditional distributions. Due to its general estimation procedure, it can be employed to estimate a wide range of targets such as conditional average treatment effects, conditional quantiles, and conditional correlations. However, only results about the consistency and convergence rate of the DRF prediction are available so far. We characterize the asymptotic distribution of DRF and develop a bootstrap approximation of it. This allows us to derive inferential tools for quantifying standard errors and the construction of confidence regions that have asymptotic coverage guarantees. In simulation studies, we empirically validate the developed theory for inference of low-dimensional targets and for testing distributional differences between two populations.

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Juan L. Gamella, Armeen Taeb, Christina Heinze-Deml, Peter Bühlmann

We consider the problem of recovering the causal structure underlying observations from different experimental conditions when the targets of the interventions in each experiment are unknown. We assume a linear structural causal model with additive Gaussian noise and consider interventions that perturb their targets while maintaining the causal relationships in the system. Different models may entail the same distributions, offering competing causal explanations for the given observations. We fully characterize this equivalence class and offer identifiability results, which we use to derive a greedy algorithm called GnIES to recover the equivalence class of the data-generating model without knowledge of the intervention targets. In addition, we develop a novel procedure to generate semi-synthetic data sets with known causal ground truth but distributions closely resembling those of a real data set of choice. We leverage this procedure and evaluate the performance of GnIES on synthetic, real, and semi-synthetic data sets. Despite the strong Gaussian distributional assumption, GnIES is robust to an array of model violations and competitive in recovering the causal structure in small- to large-sample settings. We provide, in the Python packages "gnies" and "sempler", implementations of GnIES and our semi-synthetic data generation procedure.

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Alexander Immer, Christoph Schultheiss, Julia E. Vogt, Bernhard Schölkopf, Peter Bühlmann, Alexander Marx

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 probabilistic neural networks. Both model the conditional distribution of $Y$ given $X$ as a Gaussian parameterized by its natural parameters. Since the neural network approach can fit functions of arbitrary complexity, it has an edge over the feature map-based approach in terms of empirical performance. When the feature maps are correctly specified, however, we can prove that our estimator is jointly concave, which allows us to derive stronger guarantees for the cause-effect identification task.

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