Causal reasoning is a crucial part of science and human intelligence. In order to discover causal relationships from data, we need structure discovery methods. We provide a review of background theory and a survey of methods for structure discovery. We primarily focus on modern, continuous optimization methods, and provide reference to further resources such as benchmark datasets and software packages. Finally, we discuss the assumptive leap required to take us from structure to causality.
Università Cattolica del Sacro Cuore（天主教圣心大学）；Università degli Studi di Milano-Bicocca（米兰比可卡大学）
Directed Acyclic Graphs (DAGs) provide a powerful framework to model causal relationships among variables in multivariate settings; in addition, through the do-calculus theory, they allow for the identification and estimation of causal effects between variables also from pure observational data. In this setting, the process of inferring the DAG structure from the data is referred to as causal structure learning or causal discovery. We introduce BCDAG, an R package for Bayesian causal discovery and causal effect estimation from Gaussian observational data, implementing the Markov chain Monte Ca…
Causal processes in biomedicine may contain cycles, evolve over time or differ between populations. However, many graphical models cannot accommodate these conditions. We propose to model causation using a mixture of directed cyclic graphs (DAGs), where the joint distribution in a population follows a DAG at any single point in time but potentially different DAGs across time. We also introduce an algorithm called Causal Inference over Mixtures that uses longitudinal data to infer a graph summarizing the causal relations generated from a mixture of DAGs. Experiments demonstrate improved perform…
Massachusetts Institute of Technology（麻省理工学院）；University of Michigan（密歇根大学）；ETH Zurich（苏黎世联邦理工学院）
We consider distributions arising from a mixture of causal models, where each model is represented by a directed acyclic graph (DAG). We provide a graphical representation of such mixture distributions and prove that this representation encodes the conditional independence relations of the mixture distribution. We then consider the problem of structure learning based on samples from such distributions. Since the mixing variable is latent, we consider causal structure discovery algorithms such as FCI that can deal with latent variables. We show that such algorithms recover a "union" of the comp…
Columbia University（哥伦比亚大学）；Iowa State University（爱荷华州立大学）
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 (Pearl, 2000). 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…
RAND Corporation（兰德公司）；Pardee RAND Graduate School（帕迪兰德研究生院）；University of Southern California（南加州大学）
Fuzzy cognitive maps (FCMs) model feedback causal relations in interwoven webs of causality and policy variables. FCMs are fuzzy signed directed graphs that allow degrees of causal influence and event occurrence. Such causal models can simulate a wide range of policy scenarios and decision processes. Their directed loops or cycles directly model causal feedback. Their nonlinear dynamics permit forward-chaining inference from input causes and policy options to output effects. Users can add detailed dynamics and feedback links directly to the causal model or infer them with statistical learning …
Estimate Collapsibility of Causal Effects in Completed Partial DAGs via Strong d-Convex Hulls
College of Mathematics and System Science, Xinjiang University（新疆大学数学与系统科学学院）；Institute of Statistics and Data Science, Xinjiang University of Finance and Economics（新疆财经大学统计与数据科学研究院）
Directed acyclic graphs (DAGs) are frequently used in epidemiology as a method to encode causal inference assumptions. We propose the DAGWOOD framework to bring many of those encoded assumptions to the forefront. DAGWOOD combines a root DAG (the DAG in the proposed analysis) and a set of branch DAGs (alternative hidden assumptions to the root DAG). All branch DAGs share a common ruleset, and must 1) change the root DAG, 2) be a valid DAG, and either 3a) change the minimally sufficient adjustment set or 3b) change the number of frontdoor paths. Branch DAGs comprise a list of assumptions which m…
The problem of learning a directed acyclic graph (DAG) up to Markov equivalence is equivalent to the problem of finding a permutation of the variables that induces the sparsest graph. Without additional assumptions, this task is known to be NP-hard. Building on the minimum degree algorithm for sparse Cholesky decomposition, but utilizing DAG-specific problem structure, we introduce an efficient algorithm for finding such sparse permutations. We show that on jointly Gaussian distributions, our method with depth $w$ runs in $O(p^{w+3})$ time. We compare our method with $w = 1$ to algorithms for …
Causal structures for observational survival data provide crucial information regarding the relationships between covariates and time-to-event. We derive motivation from the information theoretic source coding argument, and show that incorporating the knowledge of the directed acyclic graph (DAG) can be beneficial if suitable source encoders are employed. As a possible source encoder in this context, we derive a variational inference based conditional variational autoencoder for causal structured survival prediction, which we refer to as DAGSurv. We illustrate the performance of DAGSurv on low…
Williams College（威廉姆斯学院）；Emory University（埃默里大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Identification theory for causal effects in causal models associated with hidden variable directed acyclic graphs (DAGs) is well studied. However, the corresponding algorithms are underused due to the complexity of estimating the identifying functionals they output. In this work, we bridge the gap between identification and estimation of population-level causal effects involving a single treatment and a single outcome. We derive influence function based estimators that exhibit double robustness for the identified effects in a large class of hidden variable DAGs where the treatment satisfies a …
Stanford University（斯坦福大学）；Facebook AI Research（Facebook人工智能研究院）；University of California, Los Angeles（加州大学洛杉矶分校）
A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identifiable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizi…
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of California Los Angeles（加州大学洛杉矶分校）
Unobserved confounding is one of the greatest challenges for causal discovery. The case in which unobserved variables have a widespread effect on many of the observed ones is particularly difficult because most pairs of variables are conditionally dependent given any other subset, rendering the causal effect unidentifiable. In this paper we show that beyond conditional independencies, under the principle of independent mechanisms, unobserved confounding in this setting leaves a statistical footprint in the observed data distribution that allows for disentangling spurious and causal effects. Us…
Causal structure learning with momentum: Sampling distributions over Markov Equivalence Classes of DAGs
Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）；Institute for Theoretical Computer Science（理论计算机科学研究所）；University of Lübeck（吕贝克大学）
Deterministic variables are variables that are fully explained by one or more parent variables. They commonly arise when a variable has been algebraically constructed from one or more parent variables, as with composite variables, and in compositional data, where the 'whole' variable is determined from its 'parts'. This article introduces how deterministic variables may be depicted within directed acyclic graphs (DAGs) to help with identifying and interpreting causal effects involving tautological associations, compositional data, and composite variables. We propose a two-step approach in whic…
CaTs and DAGs: Integrating Directed Acyclic Graphs with Transformers for Causally Constrained Predictions
Kivira Health（Kivira健康公司）；The Sense Innovation and Research Center, CHUV, Lausanne, Switzerland（The Sense创新与研究中心，CHUV，洛桑，瑞士）；Centre for Vision, Speech and Signal Processing, University of Surrey, U.K.（视觉、语音和信号处理中心，萨里大学，英国）；Institute of Psychology, University of Lausanne, Switzerland（心理学研究所，洛桑大学，瑞士）；EPFL, Switzerland（瑞士联邦理工学院）
Graphical models based on Directed Acyclic Graphs (DAGs) are widely used to answer causal questions across a variety of scientific and social disciplines. However, observational data alone cannot distinguish in general between DAGs representing the same conditional independence assertions (Markov equivalent DAGs); as a consequence the orientation of some edges in the graph remains indeterminate. Interventional data, produced by exogenous manipulations of variables in the network, enhance the process of structure learning because they allow to distinguish among equivalent DAGs, thus sharpening …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Sharif University of Technology（谢里夫理工大学）；Georgia Institute of Technology（佐治亚理工学院）；Carnegie Mellon University（卡内基梅隆大学）
A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. I…
UCLA Department of Biostatistics（加州大学洛杉矶分校生物统计系）；UCLA Department of Statistics（加州大学洛杉矶分校统计系）
Discovery of causal relationships from observational data is an important problem in many areas. Several recent results have established the identifiability of causal DAGs with non-Gaussian and/or nonlinear structural equation models (SEMs). In this paper, we focus on nonlinear SEMs defined by non-invertible functions, which exist in many data domains, and propose a novel test for non-invertible bivariate causal models. We further develop a method to incorporate this test in structure learning of DAGs that contain both linear and nonlinear causal relations. By extensive numerical comparisons, …
Tags for DAGs: Graph Refinement with Meta-Informed Relations
Computer Science Department, Technical University of Darmstadt（达姆施塔特技术大学计算机科学系）；Hessian Center for AI (hessian.AI)（海德堡人工智能中心）；German Research Center for AI (DFKI)（德国人工智能研究中心）；Centre for Cognitive Science, Technical University of Darmstadt（达姆施塔特技术大学认知科学中心）；Department of Mathematics and Computer Science, Eindhoven University of Technology（埃因霍温理工大学数学与计算机科学系）
ETH Zurich（苏黎世联邦理工学院）；Mila（米拉人工智能研究所）；Université de Montréal（蒙特利尔大学）；ElementAI；ServiceNow；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；CIFAR（加拿大高等研究院）
Learning the causal structure that underlies data is a crucial step towards robust real-world decision making. The majority of existing work in causal inference focuses on determining a single directed acyclic graph (DAG) or a Markov equivalence class thereof. However, a crucial aspect to acting intelligently upon the knowledge about causal structure which has been inferred from finite data demands reasoning about its uncertainty. For instance, planning interventions to find out more about the causal mechanisms that govern our data requires quantifying epistemic uncertainty over DAGs. While Ba…
Beyond DAGs: A Latent Partial Causal Model for Multimodal Learning
Responsible AI Research Centre（负责任人工智能研究中心）；Australian Institute for Machine Learning, Adelaide University（阿德莱德大学澳大利亚机器学习研究所）；School of Computer Science and Engineering, The University of New South Wales（新南威尔士大学计算机科学与工程学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）；Department of Philosophy, Carnegie Mellon University（卡内基梅隆大学哲学系）
Real causal processes may contain feedback loops and change over time. In this paper, we model cycles and non-stationary distributions using a mixture of directed acyclic graphs (DAGs). We then study the conditional independence (CI) relations induced by a density that factorizes according to a mixture of DAGs in two steps. First, we generalize d-separation for a single DAG to mixture d-separation for a mixture of DAGs. We then utilize the mixture d-separation criterion to derive a global Markov property that allows us to read off the CI relations induced by a mixture of DAGs using a particula…
Johns Hopkins University（约翰斯·霍普金斯大学）；Perimeter Institute for Theoretical Physics（圆周理论物理研究所）；University of Gdańsk（格但斯克大学）；International Centre for Theory of Quantum Technologies（国际量子技术理论中心）
Directed acyclic graphs (DAGs) with hidden variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. In this work, we present entropic inequality constraints that are implied by $e$-separation relations in hidden variable DAGs with discrete observed variables. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entr…
Personalized Binomial DAGs Learning with Network Structured Covariates
Booth School of Business, University of Chicago（芝加哥大学布斯商学院）；University of Michigan（密歇根大学）；Ant Financial Group（蚂蚁集团）；Marshall School of Business, University of Southern California（南加州大学马歇尔商学院）
Conventional methods for causal structure learning from data face significant challenges due to combinatorial search space. Recently, the problem has been formulated into a continuous optimization framework with an acyclicity constraint to learn Directed Acyclic Graphs (DAGs). Such a framework allows the utilization of deep generative models for causal structure learning to better capture the relations between data sample distributions and DAGs. However, so far no study has experimented with the use of Wasserstein distance in the context of causal structure learning. Our model named DAG-WGAN c…
The PC and FCI algorithms are popular constraint-based methods for learning the structure of directed acyclic graphs (DAGs) in the absence and presence of latent and selection variables, respectively. These algorithms (and their order-independent variants, PC-stable and FCI-stable) have been shown to be consistent for learning sparse high-dimensional DAGs based on partial correlations. However, inferring conditional independences from partial correlations is valid if the data are jointly Gaussian or generated from a linear structural equation model -- an assumption that may be violated in many…
Background: In epidemiology, causal inference and prediction modeling methodologies have been historically distinct. Directed Acyclic Graphs (DAGs) are used to model a priori causal assumptions and inform variable selection strategies for causal questions. Although tools originally designed for prediction are finding applications in causal inference, the counterpart has remained largely unexplored. The aim of this theoretical and simulation-based study is to assess the potential benefit of using DAGs in clinical risk prediction modeling. Methods and Findings: We explore how incorporating knowl…
Stable Causal Discovery via Directed Acyclic Graph Aggregation
Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Department of Statistics, University of Virginia（弗吉尼亚大学统计系）；Dr. Bing Zhang Department of Statistics, University of Kentucky（肯塔基大学张炳章统计系）
Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly identify all arrow orientations shared by all DAGs in a Markov equivalence class, given a member of that class. For DAG models with latent variables, maximal ancestral graphs (MAGs) provide a neat representation that facilitates model search. Earlier work (Ali et al. 2005) has ide…
Models and Algorithms for Data and Text Mining Laboratory (MADLab) Department of Informatics Systems and Communication (DISCo) University of Milano - Bicocca Milan Italy（数据与文本挖掘算法实验室（MADLab）信息系统与通信系（DISCo）米兰-比科卡大学）；Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA) Lugano Switzerland（智能研究 institute（IDSIA）卢加诺瑞士）；University of Milano - Bicocca Milan Italy（米兰-比科卡大学）
In this review, we discuss approaches for learning causal structure from data, also called causal discovery. In particular, we focus on approaches for learning directed acyclic graphs (DAGs) and various generalizations which allow for some variables to be unobserved in the available data. We devote special attention to two fundamental combinatorial aspects of causal structure learning. First, we discuss the structure of the search space over causal graphs. Second, we discuss the structure of equivalence classes over causal graphs, i.e., sets of graphs which represent what can be learned from o…
