Tom Claassen

Kristian Gonzalez Barman, Sascha Caron, Tom Claassen et al.

Scientific understanding is a fundamental goal of science, allowing us to explain the world. There is currently no good way to measure the scientific understanding of agents, whether these be humans or Artificial Intelligence systems. Without a clear benchmark, it is challenging to evaluate and compare different levels of and approaches to scientific understanding. In this Roadmap, we propose a framework to create a benchmark for scientific understanding, utilizing tools from philosophy of science. We adopt a behavioral notion according to which genuine understanding should be recognized as an ability to perform certain tasks. We extend this notion by considering a set of questions that can gauge different levels of scientific understanding, covering information retrieval, the capability to arrange information to produce an explanation, and the ability to infer how things would be different under different circumstances. The Scientific Understanding Benchmark (SUB), which is formed by a set of these tests, allows for the evaluation and comparison of different approaches. Benchmarking plays a crucial role in establishing trust, ensuring quality control, and providing a basis for performance evaluation. By aligning machine and human scientific understanding we can improve their utility, ultimately advancing scientific understanding and helping to discover new insights within machines.

Mátyás Schubert, Tom Claassen, Sara Magliacane

Causal discovery can be computationally demanding for large numbers of variables. If we only wish to estimate the causal effects on a small subset of target variables, we might not need to learn the causal graph for all variables, but only a small subgraph that includes the targets and their adjustment sets. In this paper, we focus on identifying causal effects between target variables in a computationally and statistically efficient way. This task combines causal discovery and effect estimation, aligning the discovery objective with the effects to be estimated. We show that definite non-ancestors of the targets are unnecessary to learn causal relations between the targets and to identify efficient adjustments sets. We sequentially identify and prune these definite non-ancestors with our Sequential Non-Ancestor Pruning (SNAP) framework, which can be used either as a preprocessing step to standard causal discovery methods, or as a standalone sound and complete causal discovery algorithm. Our results on synthetic and real data show that both approaches substantially reduce the number of independence tests and the computation time without compromising the quality of causal effect estimations.

Mátyás Schubert, Tom Claassen, Sara Magliacane

Causal discovery methods can identify valid adjustment sets for causal effect estimation for a pair of target variables, even when the underlying causal graph is unknown. Global causal discovery methods focus on learning the whole causal graph and therefore enable the recovery of optimal adjustment sets, i.e., sets with the lowest asymptotic variance, but they quickly become computationally prohibitive as the number of variables grows. Local causal discovery methods offer a more scalable alternative by focusing on the local neighborhood of the target variables, but are restricted to statistically suboptimal adjustment sets. In this work, we propose Local Optimal Adjustments Discovery (LOAD), a sound and complete causal discovery approach that combines the computational efficiency of local methods with the statistical optimality of global methods. First, LOAD identifies the causal relation between the targets and tests if the causal effect is identifiable by using only local information. If it is identifiable, it then finds the optimal adjustment set by leveraging local causal discovery to infer the mediators and their parents. Otherwise, it returns the locally valid parent adjustment sets based on the learned local structure. In our experiments on synthetic and realistic data LOAD outperforms global methods in scalability, while providing more accurate effect estimation than local methods.

Tom Claassen, Joris M. Mooij

We present a new, efficient procedure to establish Markov equivalence between directed graphs that may or may not contain cycles under the \textit{d}-separation criterion. It is based on the Cyclic Equivalence Theorem (CET) in the seminal works on cyclic models by Thomas Richardson in the mid '90s, but now rephrased from an ancestral perspective. The resulting characterization leads to a procedure for establishing Markov equivalence between graphs that no longer requires tests for d-separation, leading to a significantly reduced algorithmic complexity. The conceptually simplified characterization may help to reinvigorate theoretical research towards sound and complete cyclic discovery in the presence of latent confounders. This version includes a correction to rule (iv) in Theorem 1, and the subsequent adjustment in part 2 of Algorithm 2.

Ioan Gabriel Bucur, Tom Claassen, Tom Heskes

The use of genetic variants as instrumental variables - an approach known as Mendelian randomization - is a popular epidemiological method for estimating the causal effect of an exposure (phenotype, biomarker, risk factor) on a disease or health-related outcome from observational data. Instrumental variables must satisfy strong, often untestable assumptions, which means that finding good genetic instruments among a large list of potential candidates is challenging. This difficulty is compounded by the fact that many genetic variants influence more than one phenotype through different causal pathways, a phenomenon called horizontal pleiotropy. This leads to errors not only in estimating the magnitude of the causal effect but also in inferring the direction of the putative causal link. In this paper, we propose a Bayesian approach called BayesMR that is a generalization of the Mendelian randomization technique in which we allow for pleiotropic effects and, crucially, for the possibility of reverse causation. The output of the method is a posterior distribution over the target causal effect, which provides an immediate and easily interpretable measure of the uncertainty in the estimation. More importantly, we use Bayesian model averaging to determine how much more likely the inferred direction is relative to the reverse direction.

Ioan Gabriel Bucur, Tom Claassen, Tom Heskes

The recent availability of huge, many-dimensional data sets, like those arising from genome-wide association studies (GWAS), provides many opportunities for strengthening causal inference. One popular approach is to utilize these many-dimensional measurements as instrumental variables (instruments) for improving the causal effect estimate between other pairs of variables. Unfortunately, searching for proper instruments in a many-dimensional set of candidates is a daunting task due to the intractable model space and the fact that we cannot directly test which of these candidates are valid, so most existing search methods either rely on overly stringent modeling assumptions or fail to capture the inherent model uncertainty in the selection process. We show that, as long as at least some of the candidates are (close to) valid, without knowing a priori which ones, they collectively still pose enough restrictions on the target interaction to obtain a reliable causal effect estimate. We propose a general and efficient causal inference algorithm that accounts for model uncertainty by performing Bayesian model averaging over the most promising many-dimensional instrumental variable models, while at the same time employing weaker assumptions regarding the data generating process. We showcase the efficiency, robustness and predictive performance of our algorithm through experimental results on both simulated and real-world data.

Tom Heskes, Evi Sijben, Ioan Gabriel Bucur et al.

Shapley values underlie one of the most popular model-agnostic methods within explainable artificial intelligence. These values are designed to attribute the difference between a model's prediction and an average baseline to the different features used as input to the model. Being based on solid game-theoretic principles, Shapley values uniquely satisfy several desirable properties, which is why they are increasingly used to explain the predictions of possibly complex and highly non-linear machine learning models. Shapley values are well calibrated to a user's intuition when features are independent, but may lead to undesirable, counterintuitive explanations when the independence assumption is violated. In this paper, we propose a novel framework for computing Shapley values that generalizes recent work that aims to circumvent the independence assumption. By employing Pearl's do-calculus, we show how these 'causal' Shapley values can be derived for general causal graphs without sacrificing any of their desirable properties. Moreover, causal Shapley values enable us to separate the contribution of direct and indirect effects. We provide a practical implementation for computing causal Shapley values based on causal chain graphs when only partial information is available and illustrate their utility on a real-world example.

Joris M. Mooij, Tom Claassen

While feedback loops are known to play important roles in many complex systems, their existence is ignored in a large part of the causal discovery literature, as systems are typically assumed to be acyclic from the outset. When applying causal discovery algorithms designed for the acyclic setting on data generated by a system that involves feedback, one would not expect to obtain correct results. In this work, we show that -- surprisingly -- the output of the Fast Causal Inference (FCI) algorithm is correct if it is applied to observational data generated by a system that involves feedback. More specifically, we prove that for observational data generated by a simple and $σ$-faithful Structural Causal Model (SCM), FCI is sound and complete, and can be used to consistently estimate (i) the presence and absence of causal relations, (ii) the presence and absence of direct causal relations, (iii) the absence of confounders, and (iv) the absence of specific cycles in the causal graph of the SCM. We extend these results to constraint-based causal discovery algorithms that exploit certain forms of background knowledge, including the causally sufficient setting (e.g., the PC algorithm) and the Joint Causal Inference setting (e.g., the FCI-JCI algorithm).

Ioan Gabriel Bucur, Tom Claassen, Tom Heskes

Gene regulatory networks play a crucial role in controlling an organism's biological processes, which is why there is significant interest in developing computational methods that are able to extract their structure from high-throughput genetic data. Many of these computational methods are designed to infer individual regulatory relationships among genes from data on gene expression. We propose a novel efficient Bayesian method for discovering local causal relationships among triplets of (normally distributed) variables. In our approach, we score covariance structures for each triplet in one go and incorporate available background knowledge in the form of priors to derive posterior probabilities over local causal structures. Our method is flexible in the sense that it allows for different types of causal structures and assumptions. We apply our approach to the task of learning causal regulatory relationships among genes. We show that the proposed algorithm produces stable and conservative posterior probability estimates over local causal structures that can be used to derive an honest ranking of the most meaningful regulatory relationships. We demonstrate the stability and efficacy of our method both on simulated data and on real-world data from an experiment on yeast.

Ioan Gabriel Bucur, Tom van Bussel, Tom Claassen et al.

Gene regulatory networks play a crucial role in controlling an organism's biological processes, which is why there is significant interest in developing computational methods that are able to extract their structure from high-throughput genetic data. A typical approach consists of a series of conditional independence tests on the covariance structure meant to progressively reduce the space of possible causal models. We propose a novel efficient Bayesian method for discovering the local causal relationships among triplets of (normally distributed) variables. In our approach, we score the patterns in the covariance matrix in one go and we incorporate the available background knowledge in the form of priors over causal structures. Our method is flexible in the sense that it allows for different types of causal structures and assumptions. We apply the approach to the task of inferring gene regulatory networks by learning regulatory relationships between gene expression levels. We show that our algorithm produces stable and conservative posterior probability estimates over local causal structures that can be used to derive an honest ranking of the most meaningful regulatory relationships. We demonstrate the stability and efficacy of our method both on simulated data and on real-world data from an experiment on yeast.

Sara Magliacane, Thijs van Ommen, Tom Claassen et al.

An important goal common to domain adaptation and causal inference is to make accurate predictions when the distributions for the source (or training) domain(s) and target (or test) domain(s) differ. In many cases, these different distributions can be modeled as different contexts of a single underlying system, in which each distribution corresponds to a different perturbation of the system, or in causal terms, an intervention. We focus on a class of such causal domain adaptation problems, where data for one or more source domains are given, and the task is to predict the distribution of a certain target variable from measurements of other variables in one or more target domains. We propose an approach for solving these problems that exploits causal inference and does not rely on prior knowledge of the causal graph, the type of interventions or the intervention targets. We demonstrate our approach by evaluating a possible implementation on simulated and real world data.

Ioan Gabriel Bucur, Tom Claassen, Tom Heskes

Causal effect estimation from observational data is an important and much studied research topic. The instrumental variable (IV) and local causal discovery (LCD) patterns are canonical examples of settings where a closed-form expression exists for the causal effect of one variable on another, given the presence of a third variable. Both rely on faithfulness to infer that the latter only influences the target effect via the cause variable. In reality, it is likely that this assumption only holds approximately and that there will be at least some form of weak interaction. This brings about the paradoxical situation that, in the large-sample limit, no predictions are made, as detecting the weak edge invalidates the setting. We introduce an alternative approach by replacing strict faithfulness with a prior that reflects the existence of many 'weak' (irrelevant) and 'strong' interactions. We obtain a posterior distribution over the target causal effect estimator which shows that, in many cases, we can still make good estimates. We demonstrate the approach in an application on a simple linear-Gaussian setting, using the MultiNest sampling algorithm, and compare it with established techniques to show our method is robust even when strict faithfulness is violated.

Joris M. Mooij, Sara Magliacane, Tom Claassen

The gold standard for discovering causal relations is by means of experimentation. Over the last decades, alternative methods have been proposed that can infer causal relations between variables from certain statistical patterns in purely observational data. We introduce Joint Causal Inference (JCI), a novel approach to causal discovery from multiple data sets from different contexts that elegantly unifies both approaches. JCI is a causal modeling framework rather than a specific algorithm, and it can be implemented using any causal discovery algorithm that can take into account certain background knowledge. JCI can deal with different types of interventions (e.g., perfect, imperfect, stochastic, etc.) in a unified fashion, and does not require knowledge of intervention targets or types in case of interventional data. We explain how several well-known causal discovery algorithms can be seen as addressing special cases of the JCI framework, and we also propose novel implementations that extend existing causal discovery methods for purely observational data to the JCI setting. We evaluate different JCI implementations on synthetic data and on flow cytometry protein expression data and conclude that JCI implementations can considerably outperform state-of-the-art causal discovery algorithms.

Sara Magliacane, Tom Claassen, Joris M. Mooij

Constraint-based causal discovery from limited data is a notoriously difficult challenge due to the many borderline independence test decisions. Several approaches to improve the reliability of the predictions by exploiting redundancy in the independence information have been proposed recently. Though promising, existing approaches can still be greatly improved in terms of accuracy and scalability. We present a novel method that reduces the combinatorial explosion of the search space by using a more coarse-grained representation of causal information, drastically reducing computation time. Additionally, we propose a method to score causal predictions based on their confidence. Crucially, our implementation also allows one to easily combine observational and interventional data and to incorporate various types of available background knowledge. We prove soundness and asymptotic consistency of our method and demonstrate that it can outperform the state-of-the-art on synthetic data, achieving a speedup of several orders of magnitude. We illustrate its practical feasibility by applying it on a challenging protein data set.

Tom Claassen, Joris M. Mooij, Tom Heskes

This article contains detailed proofs and additional examples related to the UAI-2013 submission `Learning Sparse Causal Models is not NP-hard'. It describes the FCI+ algorithm: a method for sound and complete causal model discovery in the presence of latent confounders and/or selection bias, that has worst case polynomial complexity of order $N^{2(k+1)}$ in the number of independence tests, for sparse graphs over $N$ nodes, bounded by node degree $k$. The algorithm is an adaptation of the well-known FCI algorithm by (Spirtes et al., 2000) that is also sound and complete, but has worst case complexity exponential in $N$.

Tom Claassen, Joris Mooij, Tom Heskes

This paper shows that causal model discovery is not an NP-hard problem, in the sense that for sparse graphs bounded by node degree k the sound and complete causal model can be obtained in worst case order N^{2(k+2)} independence tests, even when latent variables and selection bias may be present. We present a modification of the well-known FCI algorithm that implements the method for an independence oracle, and suggest improvements for sample/real-world data versions. It does not contradict any known hardness results, and does not solve an NP-hard problem: it just proves that sparse causal discovery is perhaps more complicated, but not as hard as learning minimal Bayesian networks.

Tom Claassen, Tom Heskes

We target the problem of accuracy and robustness in causal inference from finite data sets. Some state-of-the-art algorithms produce clear output complete with solid theoretical guarantees but are susceptible to propagating erroneous decisions, while others are very adept at handling and representing uncertainty, but need to rely on undesirable assumptions. Our aim is to combine the inherent robustness of the Bayesian approach with the theoretical strength and clarity of constraint-based methods. We use a Bayesian score to obtain probability estimates on the input statements used in a constraint-based procedure. These are subsequently processed in decreasing order of reliability, letting more reliable decisions take precedence in case of con icts, until a single output model is obtained. Tests show that a basic implementation of the resulting Bayesian Constraint-based Causal Discovery (BCCD) algorithm already outperforms established procedures such as FCI and Conservative PC. It can also indicate which causal decisions in the output have high reliability and which do not.

Tom Claassen, Tom Heskes

We present a novel approach to constraint-based causal discovery, that takes the form of straightforward logical inference, applied to a list of simple, logical statements about causal relations that are derived directly from observed (in)dependencies. It is both sound and complete, in the sense that all invariant features of the corresponding partial ancestral graph (PAG) are identified, even in the presence of latent variables and selection bias. The approach shows that every identifiable causal relation corresponds to one of just two fundamental forms. More importantly, as the basic building blocks of the method do not rely on the detailed (graphical) structure of the corresponding PAG, it opens up a range of new opportunities, including more robust inference, detailed accountability, and application to large models.