Testing a hypothesized causal model against observational data is a key prerequisite for many causal inference tasks. A natural approach is to test whether the conditional independence relations (CIs) assumed in the model hold in the data. While a model can assume exponentially many CIs (with respect to the number of variables), testing all of them is both impractical and unnecessary. Causal graphs, which encode these CIs in polynomial space, give rise to local Markov properties that enable model testing with a significantly smaller subset of CIs. Model testing based on local properties requires an algorithm to list the relevant CIs. However, existing algorithms for realistic settings with hidden variables and non-parametric distributions can take exponential time to produce even a single CI constraint. In this paper, we introduce the c-component local Markov property (C-LMP) for causal graphs with hidden variables. Since C-LMP can still invoke an exponential number of CIs, we develop a polynomial delay algorithm to list these CIs in poly-time intervals. To our knowledge, this is the first algorithm that enables poly-delay testing of CIs in causal graphs with hidden variables against arbitrary data distributions. Experiments on real-world and synthetic data demonstrate the practicality of our algorithm.

Causal discovery methods are powerful tools for uncovering the structure of relationships among variables, yet they face significant challenges in scalability and interpretability, especially in high-dimensional settings. In many domains, researchers are not only interested in causal links between individual variables, but also in relationships among sets or clusters of variables. Learning causal structure at the cluster level can both reveal higher-order relationships of interest and improve scalability. In this work, we introduce an approach for causal discovery over clusters in Markov causal systems. We propose a new graphical model that encodes knowledge of relationships between user-defined clusters while fully representing independencies and dependencies over clusters, faithful to a given distribution. We then define and characterize a graphical equivalence class of these models that share cluster-level independence information. Lastly, we present a sound and complete algorithm for causal discovery to represent learnable causal relationships between clusters of variables.

Causal effect identification and estimation are two crucial tasks in causal inference. Although causal effect identification has been theoretically resolved, many existing estimators only address a subset of scenarios, known as the sequential back-door adjustment (SBD) (Pearl and Robins, 1995) or g-formula (Robins, 1986). Recent efforts for developing general-purpose estimators with broader coverage, incorporating the front-door adjustment (FD) (Pearl, 2000) and more, lack scalability due to the high computational cost of summing over high-dimensional variables. In this paper, we introduce a novel approach that achieves broad coverage of causal estimands beyond the SBD, incorporating various sum-product functionals like the FD, while maintaining scalability -- estimated in polynomial time relative to the number of variables and samples. Specifically, we present the class of UCA for which a scalable and doubly robust estimator is developed. In particular, we illustrate the expressiveness of UCA for a wide spectrum of causal estimands (e.g., SBD, FD, and more) in causal inference. We then develop an estimator that exhibits computational efficiency and doubly robustness. The scalability and robustness of the proposed framework are verified through simulations.

Reasoning about the effect of interventions and counterfactuals is a fundamental task found throughout the data sciences. A collection of principles, algorithms, and tools has been developed for performing such tasks in the last decades. One of the pervasive requirements found throughout this literature is the articulation of assumptions, which commonly appear in the form of causal diagrams. Despite the power of this approach, there are significant settings where the knowledge necessary to specify a causal diagram over all variables is not available, particularly in complex, high-dimensional domains. In this paper, we introduce a new graphical modeling tool called cluster DAGs (for short, C-DAGs) that allows for the partial specification of relationships among variables based on limited prior knowledge, alleviating the stringent requirement of specifying a full causal diagram. A C-DAG specifies relationships between clusters of variables, while the relationships between the variables within a cluster are left unspecified, and can be seen as a graphical representation of an equivalence class of causal diagrams that share the relationships among the clusters. We develop the foundations and machinery for valid inferences over C-DAGs about the clusters of variables at each layer of Pearl's Causal Hierarchy - L1 (probabilistic), L2 (interventional), and L3 (counterfactual). In particular, we prove the soundness and completeness of d-separation for probabilistic inference in C-DAGs. Further, we demonstrate the validity of Pearl's do-calculus rules over C-DAGs and show that the standard ID identification algorithm is sound and complete to systematically compute causal effects from observational data given a C-DAG. Finally, we show that C-DAGs are valid for performing counterfactual inferences about clusters of variables.

Adversarial training (AT) methods have been found to be effective against adversarial attacks on deep neural networks. Many variants of AT have been proposed to improve its performance. Pang et al. [1] have recently shown that incorporating hypersphere embedding (HE) into the existing AT procedures enhances robustness. We observe that the existing AT procedures are not designed for the HE framework, and thus fail to adequately learn the angular discriminative information available in the HE framework. In this paper, we propose integrating HE into AT with regularization terms that exploit the rich angular information available in the HE framework. Specifically, our method, termed angular-AT, adds regularization terms to AT that explicitly enforce weight-feature compactness and inter-class separation; all expressed in terms of angular features. Experimental results show that angular-AT further improves adversarial robustness.

Adversarial training (AT) is a learning procedure that trains a deep neural network with adversary examples to improve robustness. AT and its variants are widely considered the most empirically successful against adversary examples. Along the same line, this work proposes a new training objective, PMHR-AT (Penalized Modified Huber Regularization for Adversarial training) for improving adversarial robustness. PMHR-AT minimizes both natural and adversarial risk and introduces a modified Huber loss between the natural and adversarial logits as a regularization with the regularization strength adjusted based on the similarity between the predicted natural and adversarial class probabilities. Experimental results show that the proposed method recorded a better performance than existing methods on strong attacks and offers a better trade-off between the natural accuracy and adversarial robustness.

Estimating the effects of multi-dimensional treatments (i.e., joint treatment effects) is critical in many data-intensive domains, including genetics and drug evaluation. The main challenges for studying the joint treatment effects include the need for large sample sizes to explore different treatment combinations as well as potentially unsafe treatment interactions. In this paper, we develop machinery for estimating joint treatment effects by combining data from multiple experimental datasets. In particular, first, we develop new identification conditions for determining whether a joint treatment effect can be computed in terms of multiple interventional distributions under various scenarios. Further, we develop estimators with statistically appealing properties, including consistency and robustness to model misspecification and slow convergence. Finally, we perform simulation studies, which corroborate the effectiveness of the proposed methods.

Learning cause and effect relations is arguably one of the central challenges found throughout the data sciences. Formally, determining whether a collection of observational and interventional distributions can be combined to learn a target causal relation is known as the problem of generalized identification (or g-identification) [Lee et al., 2019]. Although g-identification has been well understood and solved in theory, it turns out to be challenging to apply these results in practice, in particular when considering the estimation of the target distribution from finite samples. In this paper, we develop a new, general estimator that exhibits multiply robustness properties for g-identifiable causal functionals. Specifically, we show that any g-identifiable causal effect can be expressed as a function of generalized multi-outcome sequential back-door adjustments that are amenable to estimation. We then construct a corresponding estimator for the g-identification expression that exhibits robustness properties to bias. We analyze the asymptotic convergence properties of the estimator. Finally, we illustrate the use of the proposed estimator in experimental studies. Simulation results corroborate the theory.

Feature selection has been explored in two ways, global feature selection and instance-wise feature selection. Global feature selection picks the same feature selector for the entire dataset, while instance-wise feature selection allows different feature selectors for different data instances. We propose group-wise feature selection, a new setting that sits between global and instance-wise feature selections. In group-wise feature selection, we constrain the number of possible feature selectors to be a finite number K, which allows different feature selectors while regularizing the number of different selectors. This is for flexible trade-offs between expressiveness and model complexity. We propose two techniques to solve the problem: the first applies K-Means Clustering to the instance-wise feature selection algorithm; the second uses the mixture of experts model with Gumbel-Softmax to learn group membership and feature selector simultaneously. We evaluate our techniques and show promising results on both synthetic and real datasets.

Causal contributions measure the strengths of different causes to a target quantity. Understanding causal contributions is important in empirical sciences and data-driven disciplines since it allows to answer practical queries like “what are the contributions of each cause to the effect?” In this paper, we develop a principled method for quantifying causal contributions. First, we provide desiderata of properties axioms that causal contribution measures should satisfy and propose the do-Shapley values (inspired by do-interventions [Pearl, 2000]) as a unique method satisfying these properties. Next, we develop a criterion under which the do-Shapley values can be efficiently inferred from non-experimental data. Finally, we provide do-Shapley estimators exhibiting consistency, computational feasibility, and statistical robustness. Simulation results corroborate with the theory.

This paper investigates the problem of bounding counterfactual queries from an arbitrary collection of observational and experimental distributions and qualitative knowledge about the underlying data-generating model represented in the form of a causal diagram. We show that all counterfactual distributions in an arbitrary structural causal model (SCM) with discrete observed domains could be generated by a canonical family of SCMs with the same causal diagram where unobserved (exogenous) variables are also discrete, taking values in finite domains. Utilizing the canonical SCMs, we translate the problem of bounding counterfactuals into that of polynomial programming whose solution provides optimal bounds for the counterfactual query. Solving such polynomial programs is in general computationally expensive. We then develop effective Monte Carlo algorithms to approximate optimal bounds from a combination of observational and experimental data. Our algorithms are validated extensively on synthetic and real-world datasets.

Identifying the effects of new interventions from data is a significant challenge found across a wide range of the empirical sciences. A well-known strategy for identifying such effects is Pearl's front-door (FD) criterion. The definition of the FD criterion is declarative, only allowing one to decide whether a specific set satisfies the criterion. In this paper, we present algorithms for finding and enumerating possible sets satisfying the FD criterion in a given causal diagram. These results are useful in facilitating the practical applications of the FD criterion for causal effects estimation and helping scientists to select estimands with desired properties, e.g., based on cost, feasibility of measurement, or statistical power.

Identifying causal effects from observational data is a pervasive challenge found throughout the empirical sciences. Very general methods have been developed to decide the identifiability of a causal quantity from a combination of observational data and causal knowledge about the underlying system. In practice, however, there are still challenges to estimating identifiable causal functionals from finite samples. Recently, a method known as double/debiased machine learning (DML) (Chernozhukov et al. 2018) has been proposed to learn parameters leveraging modern machine learning techniques, which is both robust to model misspecification and bias-reducing. Still, DML has only been used for causal estimation in settings when the back-door condition (also known as conditional ignorability) holds. In this paper, we develop a new, general class of estimators for any identifiable causal functionals that exhibit DML properties, which we name DML-ID. In particular, we introduce a complete identification algorithm that returns an influence function (IF) for any identifiable causal functional. We then construct the DML estimator based on the derived IF. We show that DML-ID estimators hold the key properties of debiasedness and doubly robustness. Simulation results corroborate with the theory.

Graphical PCB design tools like KiCAD lack support for high-level abstraction such as functions and loops. To improve PCB design productivity, we hereby present BHDL, a programming framework for PCB designs. In its compact and declarative syntax, schematics and layouts can be modeled effectively and expressed concisely. Treating all circuits, even a resistor, as functions, BHDL naturally supports modularized development that builds a complex design up from smaller designs hierarchically. As an embedded Domain Specific Language (eDSL), BHDL allows users to leverage the full feature of the host language for customization and extension. Our Jupyter kernel supports web-based, REPL-style development and generates auto-placed PCBs.

General methods have been developed for estimating causal effects from observational data under causal assumptions encoded in the form of a causal graph. Most of this literature assumes that the underlying causal graph is completely specified. However, only observational data is available in most practical settings, which means that one can learn at most a Markov equivalence class (MEC) of the underlying causal graph. In this paper, we study the problem of causal estimation from a MEC represented by a partial ancestral graph (PAG), which is learnable from observational data. We develop a general estimator for any identifiable causal effects in a PAG. The result fills a gap for an end-to-end solution to causal inference from observational data to effects estimation. Specifically, we develop a complete identification algorithm that derives an influence function for any identifiable causal effects from PAGs. We then construct a double/debiased machine learning (DML) estimator that is robust to model misspecification and biases in nuisance function estimation, permitting the use of modern machine learning techniques. Simulation results corroborate with the theory.

Local treatment effects are a common quantity found throughout the empirical sciences that measure the treatment effect among those who comply with what they are assigned. Most of the literature is focused on estimating the average of such quantity, which is called the ``local average treatment effect (LATE)'' [Imbens and Angrist, 1994]). In this work, we study how to estimate the density of the local treatment effect, which is naturally more informative than its average. Specifically, we develop two families of methods for this task, namely, kernel-smoothing and model-based approaches. The kernel-smoothing-based approach estimates the density through some smooth kernel functions. The model-based approach estimates the density by projecting it onto a finite-dimensional density class. For both approaches, we derive the corresponding double/debiased machine learning-based estimators [Chernozhukov et al., 2018]. We further study the asymptotic convergence rates of the estimators and show that they are robust to the biases in nuisance function estimation. The use of the proposed methods is illustrated through both synthetic and a real dataset called 401(k).

A key challenge of big data analytics is how to collect a large volume of (labeled) data. Crowdsourcing aims to address this challenge via aggregating and estimating high-quality data (e.g., sentiment label for text) from pervasive clients/users. Existing studies on crowdsourcing focus on designing new methods to improve the aggregated data quality from unreliable/noisy clients. However, the security aspects of such crowdsourcing systems remain under-explored to date. We aim to bridge this gap in this work. Specifically, we show that crowdsourcing is vulnerable to data poisoning attacks, in which malicious clients provide carefully crafted data to corrupt the aggregated data. We formulate our proposed data poisoning attacks as an optimization problem that maximizes the error of the aggregated data. Our evaluation results on one synthetic and two real-world benchmark datasets demonstrate that the proposed attacks can substantially increase the estimation errors of the aggregated data. We also propose two defenses to reduce the impact of malicious clients. Our empirical results show that the proposed defenses can substantially reduce the estimation errors of the data poisoning attacks.

Causal effect identification is one of the most prominent and well-understood problems in causal inference. Despite the generality and power of the results developed so far, there are still challenges in their applicability to practical settings, arguably due to the finitude of the samples. Simply put, there is a gap between causal effect identification and estimation. One popular setting in which sample-efficient estimators from finite samples exist is when the celebrated back-door condition holds. In this paper, we extend weighting-based methods developed for the back-door case to more general settings, and develop novel machinery for estimating causal effects using the weighting-based method as a building block. We derive graphical criteria under which causal effects can be estimated using this new machinery and demonstrate the effectiveness of the proposed method through simulation studies.

Learning causal effects from data is a fundamental problem across the sciences. Determining the identifiability of a target effect from a combination of the observational distribution and the causal graph underlying a phenomenon is well-understood in theory. However, in practice, it remains a challenge to apply the identification theory to estimate the identified causal functionals from finite samples. Although a plethora of effective estimators have been developed under the setting known as the back-door (also called conditional ignorability), there exists still no systematic way of estimating arbitrary causal functionals that are both computationally and statistically attractive. This paper aims to bridge this gap, from causal identification to causal estimation. We note that estimating functionals from limited samples based on the empirical risk minimization (ERM) principle has been pervasive in the machine learning literature, and these methods have been extended to causal inference under the back-door setting. In this paper, we develop a learning framework that marries two families of methods, benefiting from the generality of the causal identification theory and the effectiveness of the estimators produced based on the principle of ERM. Specifically, we develop a sound and complete algorithm that generates causal functionals in the form of weighted distributions that are amenable to the ERM optimization. We then provide a practical procedure for learning causal effects from finite samples and a causal graph. Finally, experimental results support the effectiveness of our approach.

Cause-and-effect relations are one of the most valuable types of knowledge sought after throughout the data-driven sciences since they translate into stable and generalizable explanations as well as efficient and robust decision-making capabilities. Inferring these relations from data, however, is a challenging task. Two of the most common barriers to this goal are known as confounding and selection biases. The former stems from the systematic bias introduced during the treatment assignment, while the latter comes from the systematic bias during the collection of units into the sample. In this paper, we consider the problem of identifiability of causal effects when both confounding and selection biases are simultaneously present. We first investigate the problem of identifiability when all the available data is biased. We prove that the algorithm proposed by [Bareinboim and Tian, 2015] is, in fact, complete, namely, whenever the algorithm returns a failure condition, no identifiability claim about the causal relation can be made by any other method. We then generalize this setting to when, in addition to the biased data, another piece of external data is available, without bias. It may be the case that a subset of the covariates could be measured without bias (e.g., from census). We examine the problem of identifiability when a combination of biased and unbiased data is available. We propose a new algorithm that subsumes the current state-of-the-art method based on the back-door criterion.

Generalizing causal effects from a controlled experiment to settings beyond the particular study population is arguably one of the central tasks found in empirical circles. While a proper design and careful execution of the experiment would support, under mild conditions, the validity of inferences about the population in which the experiment was conducted, two challenges make the extrapolation step to different populations somewhat involved, namely, transportability and sampling selection bias. The former is concerned with disparities in the distributions and causal mechanisms between the domain (i.e., settings, population, environment) where the experiment is conducted and where the inferences are intended; the latter with distortions in the sample’s proportions due to preferential selection of units into the study. In this paper, we investigate the assumptions and machinery necessary for using covariate adjustment to correct for the biases generated by both of these problems, and generalize experimental data to infer causal effects in a new domain. We derive complete graphical conditions to determine if a set of covariates is admissible for adjustment in this new setting. Building on the graphical characterization, we develop an efficient algorithm that enumerates all possible admissible sets with poly-time delay guarantee; this can be useful for when some variables are preferred over the others due to different costs or amenability to measurement.

Selection and confounding biases are the two most common impediments to the applicability of causal inference methods in large-scale settings. We generalize the notion of backdoor adjustment to account for both biases and leverage external data that may be available without selection bias (e.g., data from census). We introduce the notion of adjustment pair and present complete graphical conditions for identifying causal effects by adjustment. We further design an algorithm for listing all admissible adjustment pairs in polynomial delay, which is useful for researchers interested in evaluating certain properties of some admissible pairs but not all (common properties include cost, variance, and feasibility to measure). Finally, we describe a statistical estimation procedure that can be performed once a set is known to be admissible, which entails different challenges in terms of finite samples.

A probabilistic query may not be estimable from observed data corrupted by missing values if the data are not missing at random (MAR). It is therefore of theoretical interest and practical importance to determine in principle whether a probabilistic query is estimable from missing data or not when the data are not MAR. We present algorithms that systematically determine whether the joint probability distribution or a target marginal distribution is estimable from observed data with missing values, assuming that the data-generation model is represented as a Bayesian network, known as m-graphs, that not only encodes the dependencies among the variables but also explicitly portrays the mechanisms responsible for the missingness process. The results significantly advance the existing work.

We study the Bayesian model averaging approach to learning Bayesian network structures (DAGs) from data. We develop new algorithms including the first algorithm that is able to efficiently sample DAGs of a moderate size (with up to about 25 variables) according to the exact structure posterior. The DAG samples can then be used to construct estimators for the posterior of any feature. We theoretically prove good properties of our estimators and empirically show that our estimators considerably outperform the estimators from the previous state- of-the-art methods.

Controlling for selection and confounding biases are two of the most challenging problems that appear in data analysis in the empirical sciences as well as in artificial intelligence tasks. The combination of previously studied methods for each of these biases in isolation is not directly applicable to certain non-trivial cases in which selection and confounding biases are simultaneously present. In this paper, we tackle these instances non-parametrically and in full generality. We provide graphical and algorithmic conditions for recoverability of interventional distributions for when selection and confounding biases are both present. Our treatment completely characterizes the class of causal effects that are recoverable in Markovian models, and is suffi- cient for Semi-Markovian models.

Ancestor relations in Bayesian networks (BNs) encode long-range causal relations among random variables. In this paper, we develop dynamic programming (DP) algorithms to compute the exact posterior probabilities of ancestor relations in Bayesian networks. Previous algorithm by Parviainen and Koivisto (2011) evaluates all possible ancestor relations in time O(n3^n) and space O(3^n). However, their algorithm assumes an order-modular prior over DAGs that does not respect Markov equivalence. The resulting posteriors would bias towards DAGs consistent with more linear orders. To adhere to the uniform prior, we develop a new DP algorithm that computes the exact posteriors of all possible ancestor relations in time O(n^25^n-1) and space O(3^n). We also discuss the extension of our algorithm to computing the posteriors of s >p >t relations, i.e., a directed path from s to t via p. We apply our algorithm to a biological data set for discovering protein signaling pathways.

The notion of missing at random (MAR) plays a central role in the theory underlying current methods for handling missing data. However the standard definition of MAR is difficult to interpret in practice. In this paper, we assume the missing data model is represented as a directed acyclic graph that not only encodes the dependencies among the variables but also explicitly portrays the causal mechanisms responsible for the missingness process. We introduce an intuitively appealing notion of MAR in such graphical models, and establish its relation with the standard MAR and a few versions of MAR used in the literature. We address the question of whether MAR is testable, given that data are corrupted by missingness, by proposing a general method for identifying testable implications imposed by the graphical structure on the observed data.

Selection bias is caused by preferential exclusion of units from the samples and represents a major obstacle to valid causal and statistical inferences; it cannot be removed by randomized experiments and can rarely be detected in either experimental or observational studies. In this paper, we provide complete graphical and algorithmic conditions for recovering conditional probabilities from selection biased data. We also provide graphical conditions for recoverability when unbiased data is available over a subset of the variables. Finally, we provide a graphical condition that generalizes the backdoor criterion and serves to recover causal effects when the data is collected under preferential selection.

In this paper we develop an algorithm to find the k-best equivalence classes of Bayesian networks. Our algorithm is capable of finding much more best DAGs than the previous algorithm that directly finds the k-best DAGs (Tian, He and Ram 2010). We demonstrate our algorithm in the task of Bayesian model averaging. Empirical results show that our algorithm significantly outperforms the k-best DAG algorithm in both time and space to achieve the same quality of approximation. Our algorithm goes beyond the maximum-a-posteriori (MAP) model by listing the most likely network structures and their relative likelihood and therefore has important applications in causal structure discovery.

In causal inference, all methods of model learning rely on testable implications, namely, properties of the joint distribution that are dictated by the model structure. These constraints, if not satisfied in the data, allow us to reject or modify the model. Most common methods of testing a linear structural equation model (SEM) rely on the likelihood ratio or chi-square test which simultaneously tests all of the restrictions implied by the model. Local constraints, on the other hand, offer increased power (Bollen and Pearl, 2013; McDonald, 2002) and, in the case of failure, provide the modeler with insight for revising the model specification. One strategy of uncovering local constraints in linear SEMs is to search for overidentified path coefficients. While these overidentifying constraints are well known, no method has been given for systematically discovering them. In this paper, we extend the half-trek criterion of (Foygel et al., 2012) to identify a larger set of structural coefficients and use it to systematically discover overidentifying constraints. Still open is the question of whether our algorithm is complete.

We address the problem of deciding whether there exists a consistent estimator of a given relation Q, when data are missing not at random. We employ a formal representation called `Missingness Graphs' to explicitly portray the causal mechanisms responsible for missingness and to encode dependencies between these mechanisms and the variables being measured. Using this representation, we define the notion of \textit{recoverability} which ensures that, for a given missingness-graph $G$ and a given query $Q$ an algorithm exists such that in the limit of large samples, it produces an estimate of $Q$ \textit{as if} no data were missing. We further present conditions that the graph should satisfy in order for recoverability to hold and devise algorithms to detect the presence of these conditions.

We perform session analysis for our domain of people search within a professional social network. We find that the content‐based method is appropriate to serve as a basis for the session identification in our domain. However, there remain some problems reported in previous research which degrade the identification performance (such as accuracy) of the content‐based method. Therefore, in this article, we propose two important refinements to address these problems. We describe the underlying rationale of our refinements and then empirically show that the content‐based method equipped with our refinements is able to achieve an excellent identification performance in our domain (such as 99.820% accuracy and 99.707% F‐measure in our experiments). Next, because the time‐based method has extremely low computation costs, which makes it suitable for many real‐world applications, we investigate the feasibility of the time‐based method in our domain by evaluating its identification performance based on our refined content‐based method. Our experiments demonstrate that the performance of the time‐based method is potentially acceptable to many real applications in our domain. Finally, we analyze several features of the identified sessions in our domain and compare them with the corresponding ones in general web search. The results illustrate the profession‐oriented characteristics of our domain.