Alexander Marx

Paweł Czyż, Frederic Grabowski, Julia E. Vogt et al. · eth-zurich

Mutual information is a general statistical dependency measure which has found applications in representation learning, causality, domain generalization and computational biology. However, mutual information estimators are typically evaluated on simple families of probability distributions, namely multivariate normal distribution and selected distributions with one-dimensional random variables. In this paper, we show how to construct a diverse family of distributions with known ground-truth mutual information and propose a language-independent benchmarking platform for mutual information estimators. We discuss the general applicability and limitations of classical and neural estimators in settings involving high dimensions, sparse interactions, long-tailed distributions, and high mutual information. Finally, we provide guidelines for practitioners on how to select appropriate estimator adapted to the difficulty of problem considered and issues one needs to consider when applying an estimator to a new data set.

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.

Paweł Czyż, Frederic Grabowski, Julia E. Vogt et al. · eth-zurich

The pointwise mutual information profile, or simply profile, is the distribution of pointwise mutual information for a given pair of random variables. One of its important properties is that its expected value is precisely the mutual information between these random variables. In this paper, we analytically describe the profiles of multivariate normal distributions and introduce a novel family of distributions, Bend and Mix Models, for which the profile can be accurately estimated using Monte Carlo methods. We then show how Bend and Mix Models can be used to study the limitations of existing mutual information estimators, investigate the behavior of neural critics used in variational estimators, and understand the effect of experimental outliers on mutual information estimation. Finally, we show how Bend and Mix Models can be used to obtain model-based Bayesian estimates of mutual information, suitable for problems with available domain expertise in which uncertainty quantification is necessary.

Imant Daunhawer, Alice Bizeul, Emanuele Palumbo et al.

Contrastive learning is a cornerstone underlying recent progress in multi-view and multimodal learning, e.g., in representation learning with image/caption pairs. While its effectiveness is not yet fully understood, a line of recent work reveals that contrastive learning can invert the data generating process and recover ground truth latent factors shared between views. In this work, we present new identifiability results for multimodal contrastive learning, showing that it is possible to recover shared factors in a more general setup than the multi-view setting studied previously. Specifically, we distinguish between the multi-view setting with one generative mechanism (e.g., multiple cameras of the same type) and the multimodal setting that is characterized by distinct mechanisms (e.g., cameras and microphones). Our work generalizes previous identifiability results by redefining the generative process in terms of distinct mechanisms with modality-specific latent variables. We prove that contrastive learning can block-identify latent factors shared between modalities, even when there are nontrivial dependencies between factors. We empirically verify our identifiability results with numerical simulations and corroborate our findings on a complex multimodal dataset of image/text pairs. Zooming out, our work provides a theoretical basis for multimodal representation learning and explains in which settings multimodal contrastive learning can be effective in practice.

Mirco Mutti, Riccardo De Santi, Marcello Restelli et al.

Posterior sampling allows exploitation of prior knowledge on the environment's transition dynamics to improve the sample efficiency of reinforcement learning. The prior is typically specified as a class of parametric distributions, the design of which can be cumbersome in practice, often resulting in the choice of uninformative priors. In this work, we propose a novel posterior sampling approach in which the prior is given as a (partial) causal graph over the environment's variables. The latter is often more natural to design, such as listing known causal dependencies between biometric features in a medical treatment study. Specifically, we propose a hierarchical Bayesian procedure, called C-PSRL, simultaneously learning the full causal graph at the higher level and the parameters of the resulting factored dynamics at the lower level. We provide an analysis of the Bayesian regret of C-PSRL that explicitly connects the regret rate with the degree of prior knowledge. Our numerical evaluation conducted in illustrative domains confirms that C-PSRL strongly improves the efficiency of posterior sampling with an uninformative prior while performing close to posterior sampling with the full causal graph.

Ina Dormuth, Sven Franke, Marlies Hafer et al.

In this study, we examine the reliability of AI-based Voting Advice Applications (VAAs) and large language models (LLMs) in providing objective political information. Our analysis is based upon a comparison with party responses to 38 statements of the Wahl-O-Mat, a well-established German online tool that helps inform voters by comparing their views with political party positions. For the LLMs, we identify significant biases. They exhibit a strong alignment (over 75% on average) with left-wing parties and a substantially lower alignment with center-right (smaller 50%) and right-wing parties (around 30%). Furthermore, for the VAAs, intended to objectively inform voters, we found substantial deviations from the parties' stated positions in Wahl-O-Mat: While one VAA deviated in 25% of cases, another VAA showed deviations in more than 50% of cases. For the latter, we even observed that simple prompt injections led to severe hallucinations, including false claims such as non-existent connections between political parties and right-wing extremist ties.

Daniel Klippert, Alexander Marx

To distinguish Markov equivalent graphs in causal discovery, it is necessary to restrict the structural causal model. Crucially, we need to be able to distinguish cause $X$ from effect $Y$ in bivariate models, that is, distinguish the two graphs $X \to Y$ and $Y \to X$. Location-scale noise models (LSNMs), in which the effect $Y$ is modeled based on the cause $X$ as $Y = f(X) + g(X)N$, form a flexible class of models that is general and identifiable in most cases. Estimating these models for arbitrary noise terms $N$, however, is challenging. Therefore, practical estimators are typically restricted to symmetric distributions, such as the normal distribution. As we showcase in this paper, when $N$ is a skewed random variable, which is likely in real-world domains, the reliability of these approaches decreases. To approach this limitation, we propose SkewD, a likelihood-based algorithm for bivariate causal discovery under LSNMs with skewed noise distributions. SkewD extends the usual normal-distribution framework to the skew-normal setting, enabling reliable inference under symmetric and skewed noise. For parameter estimation, we employ a combination of a heuristic search and an expectation conditional maximization algorithm. We evaluate SkewD on novel synthetically generated datasets with skewed noise as well as established benchmark datasets. Throughout our experiments, SkewD exhibits a strong performance and, in comparison to prior work, remains robust under high skewness.

Marlies Hafer, Alexander Marx

We consider the problem of estimating the expected causal effect $E[Y|do(X)]$ for a target variable $Y$ when treatment $X$ is set by intervention, focusing on continuous random variables. In settings without selection bias or confounding, $E[Y|do(X)] = E[Y|X]$, which can be estimated using standard regression methods. However, regression fails when systematic missingness induced by selection bias, or confounding distorts the data. Boeken et al. [2023] show that when training data is subject to selection, proxy variables unaffected by this process can, under certain constraints, be used to correct for selection bias to estimate $E[Y|X]$, and hence $E[Y|do(X)]$, reliably. When data is additionally affected by confounding, however, this equality is no longer valid. Building on these results, we consider a more general setting and propose a framework that incorporates both selection bias and confounding. Specifically, we derive theoretical conditions ensuring identifiability and recoverability of causal effects under access to external data and proxy variables. We further introduce a two-step regression estimator (TSR), capable of exploiting proxy variables to adjust for selection bias while accounting for confounding. We show that TSR coincides with prior work if confounding is absent, but achieves a lower variance. Extensive simulation studies validate TSR's correctness for scenarios which may include both selection bias and confounding with proxy variables.

Alexander Marx, Arthur Gretton, Joris M. Mooij

One of the core assumptions in causal discovery is the faithfulness assumption, i.e., assuming that independencies found in the data are due to separations in the true causal graph. This assumption can, however, be violated in many ways, including xor connections, deterministic functions or cancelling paths. In this work, we propose a weaker assumption that we call $2$-adjacency faithfulness. In contrast to adjacency faithfulness, which assumes that there is no conditional independence between each pair of variables that are connected in the causal graph, we only require no conditional independence between a node and a subset of its Markov blanket that can contain up to two nodes. Equivalently, we adapt orientation faithfulness to this setting. We further propose a sound orientation rule for causal discovery that applies under weaker assumptions. As a proof of concept, we derive a modified Grow and Shrink algorithm that recovers the Markov blanket of a target node and prove its correctness under strictly weaker assumptions than the standard faithfulness assumption.

Alexander Marx, Jilles Vreeken

Testing for conditional independence is a core aspect of constraint-based causal discovery. Although commonly used tests are perfect in theory, they often fail to reject independence in practice, especially when conditioning on multiple variables. We focus on discrete data and propose a new test based on the notion of algorithmic independence that we instantiate using stochastic complexity. Amongst others, we show that our proposed test, SCI, is an asymptotically unbiased as well as $L_2$ consistent estimator for conditional mutual information (CMI). Further, we show that SCI can be reformulated to find a sensible threshold for CMI that works well on limited samples. Empirical evaluation shows that SCI has a lower type II error than commonly used tests. As a result, we obtain a higher recall when we use SCI in causal discovery algorithms, without compromising the precision.

Alexander Marx, Jilles Vreeken

We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic information theory we can obtain SCI, a highly robust, effective and computationally efficient test for conditional independence---and show it outperforms the state of the art when applied in constraint-based inference methods such as stable PC. Second, building upon on SCI, we show how to tell apart the parents and children of a given node based on the algorithmic Markov condition. We give the Climb algorithm to efficiently discover the directed, causal Markov blanket---and show it is at least as accurate as inferring the global network, while being much more efficient. Last, but not least, we detail how we can use the Climb score to direct those edges that state of the art causal discovery algorithms based on PC or GES leave undirected---and show this improves their precision, recall and F1 scores by up to 20%.

Alexander Marx, Jilles Vreeken

We consider the fundamental problem of inferring the causal direction between two univariate numeric random variables $X$ and $Y$ from observational data. The two-variable case is especially difficult to solve since it is not possible to use standard conditional independence tests between the variables. To tackle this problem, we follow an information theoretic approach based on Kolmogorov complexity and use the Minimum Description Length (MDL) principle to provide a practical solution. In particular, we propose a compression scheme to encode local and global functional relations using MDL-based regression. We infer $X$ causes $Y$ in case it is shorter to describe $Y$ as a function of $X$ than the inverse direction. In addition, we introduce Slope, an efficient linear-time algorithm that through thorough empirical evaluation on both synthetic and real world data we show outperforms the state of the art by a wide margin.

Alexander Marx, Jilles Vreeken

Given data over the joint distribution of two random variables $X$ and $Y$, we consider the problem of inferring the most likely causal direction between $X$ and $Y$. In particular, we consider the general case where both $X$ and $Y$ may be univariate or multivariate, and of the same or mixed data types. We take an information theoretic approach, based on Kolmogorov complexity, from which it follows that first describing the data over cause and then that of effect given cause is shorter than the reverse direction. The ideal score is not computable, but can be approximated through the Minimum Description Length (MDL) principle. Based on MDL, we propose two scores, one for when both $X$ and $Y$ are of the same single data type, and one for when they are mixed-type. We model dependencies between $X$ and $Y$ using classification and regression trees. As inferring the optimal model is NP-hard, we propose Crack, a fast greedy algorithm to determine the most likely causal direction directly from the data. Empirical evaluation on a wide range of data shows that Crack reliably, and with high accuracy, infers the correct causal direction on both univariate and multivariate cause-effect pairs over both single and mixed-type data.