Abstract 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. Proxy variables unaffected by the selection process can, under certain constraints, be used to correct for selection bias to recover $$E[Y|X]$$ , and hence $$E[Y|do(X)]$$ , reliably. When data is additionally affected by confounding, recovering the causal effect from selection-biased data is more challenging and requires access to proxies to both correct for confounding and for the selection mechanism. Assuming access to such proxies from external unbiased observational data, we derive theoretical conditions ensuring identifiability and recoverability of causal effects. We further introduce a linear two-step regression estimator (TSR), which can be extended through adding non-linear basis functions, capable of exploiting proxy variables to adjust for selection bias while accounting for confounding. We show that TSR is consistent with previous estimators when confounding is absent, but achieves a lower variance. Extensive simulation studies validate TSR’s correctness for scenarios that include both selection bias and confounding with proxy variables.

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.

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.

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.

Flexibly quantifying both irreducible aleatoric and model-dependent epistemic uncertainties plays an important role for complex regression problems. While deep neural networks in principle can provide this flexibility and learn heteroscedastic aleatoric uncertainties through non-linear functions, recent works highlight that maximizing the log likelihood objective parameterized by mean and variance can lead to compromised mean fits since the gradient are scaled by the predictive variance, and propose adjustments in line with this premise. We instead propose to use the natural parametrization of the Gaussian, which has been shown to be more stable for heteroscedastic regression based on non-linear feature maps and Gaussian processes. Further, we emphasize the significance of principled regularization of the network parameters and prediction. We therefore propose an efficient Laplace approximation for heteroscedastic neural networks that allows automatic regularization through empirical Bayes and provides epistemic uncertainties, both of which improve generalization. We showcase on a range of regression problems—including a new heteroscedastic image regression benchmark—that our methods are scalable, improve over previous approaches for heteroscedastic regression, and provide epistemic uncertainty without requiring hyperparameter tuning.

We study the problem of identifying cause and effect over two univariate continuous variables $X$ and $Y$ from a sample of their joint distribution. Our focus lies on the setting when the variance of the noise may be dependent on the cause. We propose to partition the domain of the cause into multiple segments where the noise indeed is dependent. To this end, we minimize a scale-invariant, penalized regression score, finding the optimal partitioning using dynamic programming. We show under which conditions this allows us to identify the causal direction for the linear setting with heteroscedastic noise, for the non-linear setting with homoscedastic noise, as well as empirically confirm that these results generalize to the non-linear and heteroscedastic case. Altogether, the ability to model heteroscedasticity translates into an improved performance in telling cause from effect on a wide range of synthetic and real-world datasets.

Estimating mutual information (MI) between two continuous random variables X and Y allows to capture non-linear dependencies between them, non-parametrically. As such, MI estimation lies at the core of many data science applications. Yet, robustly estimating MI for high-dimensional X and Y is still an open research question. In this paper, we formulate this problem through the lens of manifold learning. That is, we leverage the common assumption that the information of X and Y is captured by a low-dimensional manifold embedded in the observed high-dimensional space and transfer it to MI estimation. As an extension to state-of-the-art kNN estimators, we propose to determine the k-nearest neighbors via geodesic distances on this manifold rather than from the ambient space, which allows us to estimate MI even in the high-dimensional setting. An empirical evaluation of our method, G-KSG, against the state-of-the-art shows that it yields good estimations of MI in classical benchmark and manifold tasks, even for high dimensional datasets, which none of the existing methods can provide.

We study the problem of inferring causal graphs from observational data. We are particularly interested in discovering graphs where all edges are oriented, as opposed to the partially directed graph that the state of the art discover. To this end, we base our approach on the algorithmic Markov condition. Unlike the statistical Markov condition, it uniquely identifies the true causal network as the one that provides the simplest— as measured in Kolmogorov complexity—factorization of the joint distribution. Although Kolmogorov complexity is not computable, we can approximate it from above via the Minimum Description Length principle, which allows us to define a consistent and computable score based on non-parametric multivariate regression. To efficiently discover causal networks in practice, we introduce the GLOBE algorithm, which greedily adds, removes, and orients edges such that it minimizes the overall cost. Through an extensive set of experiments, we show GLOBE performs very well in practice, beating the state of the art by a margin.

information for discrete-continuous mixtures using multi-dimensional adaptive histograms.

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.

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 L2 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.

We consider the problem of telling apart cause from effect between two univariate continuous-valued random variables X and Y. In general, it is impossible to make definite statements about causality without making assumptions on the underlying model; one of the most important aspects of causal inference is hence to determine under which assumptions are we able to do so. In this paper we show under which general conditions we can identify cause from effect by simply choosing the direction with the best regression score. We define a general framework of identifiable regression-based scoring functions, and show how to instantiate it in practice using regression splines. Compared to existing methods that either give strong guarantees, but are hardly applicable in practice, or provide no guarantees, but do work well in practice, our instantiation combines the best of both worlds; it gives guarantees, while empirical evaluation on synthetic and real-world data shows that it performs at least as well as the state of the art.

We consider the problem of inferring the causal direction between two univariate numeric random variables X and Y from observational data. This case is especially challenging as the graph X causes Y is Markov equivalent to the graph Y causes X , and hence it is impossible to determine the correct direction using conditional independence tests. To tackle this problem, we follow an information theoretic approach based on the algorithmic Markov condition. This postulate states that in terms of Kolmogorov complexity the factorization given by the true causal model is the most succinct description of the joint distribution. This means that we can infer that X is a likely cause of Y when we need fewer bits to first transmit the data over X , and then the data of Y as a function of X , than for the inverse direction. That is, in this paper we perform causal inference by compression. To put this notion to practice, we employ the Minimum Description Length principle, and propose a score to determine how many bits we need to transmit the data using a class of regression functions that can model both local and global functional relations. To determine whether an inference, i.e. the difference in compressed sizes, is significant, we propose two analytical significance tests based on the no-hypercompression inequality. Last, but not least, we introduce the linear-time Slope and Sloper algorithms that through thorough empirical evaluation we show outperform the state of the art by a wide margin.

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.