Graph Neural Networks (GNNs) are a powerful tool for graph representation learning and have been proven to excel in practical complex problems like neural machine translation [1], traffic forecasting [5, 47], or drug discovery [11]. In this work, we investigate to which extent the inductive bias of GNNs-encoding the causal graph information-can be exploited to answer interventional and counterfactual queries. More specifically, to approximate the interventional and counterfactual distributions induced by interventions on a casual model. To this end, we assume i) causal sufficiency-i.e., absence of hidden confounders; and, ii) access to observational data and the true causal graph. We stress that the causal graph can often be inferred from expert knowledge [52] or via one of the approaches for causal discovery [12, 42]. With this analysis we aim to complement the concurrent line of research that theoretically studies the use of Neural Networks (NN) [45], and more recently GNNs [49], for causal inference. To this end, we describe the architectural design conditions that a variational graph autoencoder (VGAE)-as a density estimator that leverages a priori graph structure-must fulfill so that it can approximate causal interventions (do-operator) and abduction-action-prediction steps [33]. The resulting Variational Causal Graph Autoencoder, referred to as VACA, enables approximating the observational, interventional and counterfactual distributions induced by a causal model with unknown structural equations. We remark that parametric assumptions on the structural causal equations are in general not testable, may thus not hold in practice [34] and may lead to inaccurate results, if misspecified.

Algorithmic decision systems are increasingly used in areas such as hiring, school admission, or loan approval. Typically, these systems rely on labeled data for training a classification model. However, in many scenarios, ground-truth labels are unavailable, and instead we have only access to imperfect labels as the result of (potentially biased) human-made decisions. Despite being imperfect, historical decisions often contain some useful information on the unobserved true labels. In this paper, we focus on scenarios where only imperfect labels are available and propose a new fair ranking-based decision system, as an alternative to traditional classification algorithms. Our approach is both intuitive and easy to implement, and thus particularly suitable for adoption in real-world settings. More in detail, we introduce a distance-based decision criterion, which incorporates useful information from historical decisions and accounts for unwanted correlation between protected and legitimate features. Through extensive experiments on synthetic and real-world data, we show that our method is fair, as it a) assigns the desirable outcome to the most qualified individuals, and b) removes the effect of stereotypes in decision-making, thereby outperforming traditional classification algorithms. Additionally, we are able to show theoretically that our method is consistent with a prominent concept of individual fairness which states that "similar individuals should be treated similarly."

While many recent works have studied the problem of algorithmic fairness from the perspective of predictions, here we investigate the fairness of recourse actions recommended to individuals to recover from an unfavourable classification. To this end, we propose two new fairness criteria at the group and individual level which---unlike prior work on equalising the average distance from the decision boundary across protected groups---are based on a causal framework that explicitly models relationships between input features, thereby allowing to capture downstream effects of recourse actions performed in the physical world. We explore how our criteria relate to others, such as counterfactual fairness, and show that fairness of recourse is complementary to fairness of prediction. We then investigate how to enforce fair recourse in the training of the classifier. Finally, we discuss whether fairness violations in the data generating process revealed by our criteria may be better addressed by societal interventions and structural changes to the system, as opposed to constraints on the classifier.

Counterfactual explanations (CFE) are being widely used to explain algorithmic decisions, especially in consequential decision-making contexts (e.g., loan approval or pretrial bail). In this context, CFEs aim to provide individuals affected by an algorithmic decision with the most similar individual (i.e., nearest individual) with a different outcome. However, while an increasing number of works propose algorithms to compute CFEs, such approaches either lack in optimality of distance (i.e., they do not return the nearest individual) and perfect coverage (i.e., they do not provide a CFE for all individuals); or they cannot handle complex models, such as neural networks. In this work, we provide a framework based on Mixed-Integer Programming (MIP) to compute nearest counterfactual explanations with provable guarantees and with runtimes comparable to gradient-based approaches. Our experiments on the Adult, COMPAS, and Credit datasets show that, in contrast with previous methods, our approach allows for efficiently computing diverse CFEs with both distance guarantees and perfect coverage.

Machine learning is increasingly used to inform decision-making in sensitive situations where decisions have consequential effects on individuals' lives. In these settings, in addition to requiring models to be accurate and robust, socially relevant values such as fairness, privacy, accountability, and explainability play an important role for the adoption and impact of said technologies. In this work, we focus on algorithmic recourse, which is concerned with providing explanations and recommendations to individuals who are unfavourably treated by automated decision-making systems. We first perform an extensive literature review, and align the efforts of many authors by presenting unified definitions, formulations, and solutions to recourse. Then, we provide an overview of the prospective research directions towards which the community may engage, challenging existing assumptions and making explicit connections to other ethical challenges such as security, privacy, and fairness.

Recent work has discussed the limitations of counterfactual explanations to recommend actions for algorithmic recourse, and argued for the need of taking causal relationships between features into consideration. Unfortunately, in practice, the true underlying structural causal model is generally unknown. In this work, we first show that it is impossible to guarantee recourse without access to the true structural equations. To address this limitation, we propose two probabilistic approaches to select optimal actions that achieve recourse with high probability given limited causal knowledge (e.g., only the causal graph). The first captures uncertainty over structural equations under additive Gaussian noise, and uses Bayesian model averaging to estimate the counterfactual distribution. The second removes any assumptions on the structural equations by instead computing the average effect of recourse actions on individuals similar to the person who seeks recourse, leading to a novel subpopulation-based interventional notion of recourse. We then derive a gradient-based procedure for selecting optimal recourse actions, and empirically show that the proposed approaches lead to more reliable recommendations under imperfect causal knowledge than non-probabilistic baselines.

Predictive models are being increasingly used to support consequential decision making at the individual level in contexts such as pretrial bail and loan approval. As a result, there is increasing social and legal pressure to provide explanations that help the affected individuals not only to understand why a prediction was output, but also how to act to obtain a desired outcome. To this end, several works have proposed methods to generate counterfactual explanations. However, they are often restricted to a particular subset of models (e.g., decision trees or linear models), and cannot directly handle the mixed (numerical and nominal) nature of the features describing each individual. In this paper, we propose a model-agnostic algorithm to generate counterfactual explanations that builds on the standard theory and tools from formal verification. Specifically, our algorithm solves a sequence of satisfiability problems, where a wide variety of predictive models and distances in mixed feature spaces, as well as natural notions of plausibility and diversity, are represented as logic formulas. Our experiments on real-world data demonstrate that our approach can flexibly handle widely deployed predictive models, while providing meaningfully closer counterfactuals than existing approaches.

Making sense of a dataset in an automatic and unsupervised fashion is a challenging problem in statistics and AI. Classical approaches for {exploratory data analysis} are usually not flexible enough to deal with the uncertainty inherent to real-world data: they are often restricted to fixed latent interaction models and homogeneous likelihoods; they are sensitive to missing, corrupt and anomalous data; moreover, their expressiveness generally comes at the price of intractable inference. As a result, supervision from statisticians is usually needed to find the right model for the data. However, since domain experts are not necessarily also experts in statistics, we propose Automatic Bayesian Density Analysis (ABDA) to make exploratory data analysis accessible at large. Specifically, ABDA allows for automatic and efficient missing value estimation, statistical data type and likelihood discovery, anomaly detection and dependency structure mining, on top of providing accurate density estimation. Extensive empirical evidence shows that ABDA is a suitable tool for automatic exploratory analysis of mixed continuous and discrete tabular data.

Consequential decisions are increasingly informed by sophisticated data-driven predictive models. For accurate predictive models, deterministic threshold rules have been shown to be optimal in terms of utility, even under a variety of fairness constraints. However, consistently learning accurate models requires access to ground truth data. Unfortunately, in practice, some data can only be observed if a certain decision was taken. Thus, collected data always depends on potentially imperfect historical decision policies. As a result, learned deterministic threshold rules are often suboptimal. We address the above question from the perspective of sequential policy learning. We first show that, if decisions are taken by a faulty deterministic policy, the observed outcomes under this policy are insufficient to improve it. We then describe how this undesirable behavior can be avoided using stochastic policies. Finally, we introduce a practical gradient-based algorithm to learn stochastic policies that effectively leverage the outcomes of decisions to improve over time. Experiments on both synthetic and real-world data illustrate our theoretical results and show the efficacy of our proposed algorithm.

Approximating a probability density in a tractable manner is a central task in Bayesian statistics. Variational Inference (VI) is a popular technique that achieves tractability by choosing a relatively simple variational approximation. Borrowing ideas from the classic boosting framework, recent approaches attempt to \emph{boost} VI by replacing the selection of a single density with an iteratively constructed mixture of densities. In order to guarantee convergence, previous works impose stringent assumptions that require significant effort for practitioners. Specifically, they require a custom implementation of the greedy step (called the LMO) for every probabilistic model with respect to an unnatural variational family of truncated distributions. Our work fixes these issues with novel theoretical and algorithmic insights. On the theoretical side, we show that boosting VI satisfies a relaxed smoothness assumption which is sufficient for the convergence of the functional Frank-Wolfe (FW) algorithm. Furthermore, we rephrase the LMO problem and propose to maximize the Residual ELBO (RELBO) which replaces the standard ELBO optimization in VI. These theoretical enhancements allow for black box implementation of the boosting subroutine. Finally, we present a stopping criterion drawn from the duality gap in the classic FW analyses and exhaustive experiments to illustrate the usefulness of our theoretical and algorithmic contributions.