Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables
Sharif University of Technology（谢里夫理工大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Georgia Institute of Technology（佐治亚理工学院）；Carnegie Mellon University（卡内基梅隆大学）
Saber Salehkaleybar, AmirEmad Ghassami, Negar Kiyavash, Kun Zhang
We consider the problem of learning causal models from observational data generated by linear non-Gaussian acyclic causal models with latent variables. Without considering the effect of latent variables, one usually infers wrong causal relationships among the observed variables. Under faithfulness assumption, we propose a method to check whether there exists a causal path between any two observed variables. From this information, we can obtain the causal order among them. The next question is then whether or not the causal effects can be uniquely identified as well. It can be shown that causal effects among observed variables cannot be identified uniquely even under the assumptions of faithfulness and non-Gaussianity of exogenous noises. However, we will propose an efficient method to identify the set of all possible causal effects that are compatible with the observational data. Furthermore, we present some structural conditions on the causal graph under which we can learn causal effects among observed variables uniquely. We also provide necessary and sufficient graphical conditions for unique identification of the number of variables in the system. Experiments on synthetic data and real-world data show the effectiveness of our proposed algorithm on learning causal models.
Causal discovery from data affected by latent confounders is an important and difficult challenge. Causal functional model-based approaches have not been used to present variables whose relationships are affected by latent confounders, while some constraint-based methods can present them. This paper proposes a causal functional model-based method called repetitive causal discovery (RCD) to discover the causal structure of observed variables affected by latent confounders. RCD repeats inferring the causal directions between a small number of observed variables and determines whether the relatio…
Responsible AI Research Centre（负责任人工智能研究中心）；Australian Institute for Machine Learning（澳大利亚机器学习研究所）；School of Computer Science and Engineering（计算机科学与工程学院）；School of Mathematics and Statistics（数学与统计学学院）；Halicioğlu Data Science Institute（Halicioğlu数据科学研究所）；Department of Philosophy（哲学系）
Spotify（声田）；University College London（伦敦大学学院）；Babylon Health（巴比伦健康）
The discovery of causal relationships is a fundamental problem in science and medicine. In recent years, many elegant approaches to discovering causal relationships between two variables from observational data have been proposed. However, most of these deal only with purely directed causal relationships and cannot detect latent common causes. Here, we devise a general heuristic which takes a causal discovery algorithm that can only distinguish purely directed causal relations and modifies it to also detect latent common causes. We apply our method to two directed causal discovery algorithms, …
Guangdong University of Technology（广东工业大学）；Peking University（北京大学）；Pazhou Lab（琶洲实验室）；Carnegie Mellon University（卡内基梅隆大学）；Foshan University（佛山大学）
Causal discovery aims to recover causal structures or models underlying the observed data. Despite its success in certain domains, most existing methods focus on causal relations between observed variables, while in many scenarios the observed ones may not be the underlying causal variables (e.g., image pixels), but are generated by latent causal variables or confounders that are causally related. To this end, in this paper, we consider Linear, Non-Gaussian Latent variable Models (LiNGLaMs), in which latent confounders are also causally related, and propose a Generalized Independent Noise (GIN…
Guangdong University of Technology（广东工业大学）；RIKEN（理化学研究所）；Shiga University（滋贺大学）；Pazhou Lab（琶洲实验室）；Peking University（北京大学）；Okayama University（冈山大学）；Foshan University（佛山科学技术学院）
Discovering causal structures among latent factors from observed data is a particularly challenging problem. Despite some efforts for this problem, existing methods focus on the single-domain data only. In this paper, we propose Multi-Domain Linear Non-Gaussian Acyclic Models for Latent Factors (MD-LiNA), where the causal structure among latent factors of interest is shared for all domains, and we provide its identification results. The model enriches the causal representation for multi-domain data. We propose an integrated two-phase algorithm to estimate the model. In particular, we first loc…
Score-Based Causal Discovery of Latent Variable Causal Models
Carnegie Mellon University（卡内基梅隆大学）；University of California, San Diego（加州大学圣地亚哥分校）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
Max Planck Institute for Biological Cybernetics（马克斯·普朗克生物控制论研究所）
In nonlinear latent variable models or dynamic models, if we consider the latent variables as confounders (common causes), the noise dependencies imply further relations between the observed variables. Such models are then closely related to causal discovery in the presence of nonlinear confounders, which is a challenging problem. However, generally in such models the observation noise is assumed to be independent across data dimensions, and consequently the noise dependencies are ignored. In this paper we focus on the Gaussian process latent variable model (GPLVM), from which we develop an ex…
University of Vienna（维也纳大学）；Vienna Cognitive Science Hub（维也纳认知科学中心）
We consider the task of causal structure learning over measurement dependence inducing latent (MeDIL) causal models. We show that this task can be framed in terms of the graph theoretic problem of finding edge clique covers,resulting in an algorithm for returning minimal MeDIL causal models (minMCMs). This algorithm is non-parametric, requiring no assumptions about linearity or Gaussianity. Furthermore, despite rather weak assumptions aboutthe class of MeDIL causal models, we show that minimality in minMCMs implies some rather specific and interesting properties. By establishing MeDIL causal m…
Universitas Islam Indonesia（印度尼西亚伊斯兰大学）；Radboud University Nijmegen（奈梅亨拉德堡德大学）
In our previous study, we introduced stable specification search for cross-sectional data (S3C). It is an exploratory causal method that combines stability selection concept and multi-objective optimization to search for stable and parsimonious causal structures across the entire range of model complexities. In this study, we extended S3C to S3C-Latent, to model causal relations between latent variables. We evaluated S3C-Latent on simulated data and compared the results to those of PC-MIMBuild, an extension of the PC algorithm, the state-of-the-art causal discovery method. The comparison showe…
University of Amsterdam（阿姆斯特丹大学）；TNO Intelligent Imaging（荷兰应用科学研究组织智能成像）；New York University（纽约大学）；CIMS（库朗数学科学研究所）；Massachusetts Institute of Technology（麻省理工学院）；CSAIL & IMES（计算机科学与人工智能实验室与医学工程与科学研究所）；University of Toronto（多伦多大学）；CIFAR（加拿大高等研究院）
Learning individual-level causal effects from observational data, such as inferring the most effective medication for a specific patient, is a problem of growing importance for policy makers. The most important aspect of inferring causal effects from observational data is the handling of confounders, factors that affect both an intervention and its outcome. A carefully designed observational study attempts to measure all important confounders. However, even if one does not have direct access to all confounders, there may exist noisy and uncertain measurement of proxies for confounders. We buil…
New York University（纽约大学）；Columbia University（哥伦比亚大学）
Causal inference from observational data requires assumptions. These assumptions range from measuring confounders to identifying instruments. Traditionally, causal inference assumptions have focused on estimation of effects for a single treatment. In this work, we construct techniques for estimation with multiple treatments in the presence of unobserved confounding. We develop two assumptions based on shared confounding between treatments and independence of treatments given the confounder. Together, these assumptions lead to a confounder estimator regularized by mutual information. For this e…
Causal processes in nature may contain cycles, and real datasets may violate causal sufficiency as well as contain selection bias. No constraint-based causal discovery algorithm can currently handle cycles, latent variables and selection bias (CLS) simultaneously. I therefore introduce an algorithm called Cyclic Causal Inference (CCI) that makes sound inferences with a conditional independence oracle under CLS, provided that we can represent the cyclic causal process as a non-recursive linear structural equation model with independent errors. Empirical results show that CCI outperforms CCD in …
Our goal is to recover time-delayed latent causal variables and identify their relations from measured temporal data. Estimating causally-related latent variables from observations is particularly challenging as the latent variables are not uniquely recoverable in the most general case. In this work, we consider both a nonparametric, nonstationary setting and a parametric setting for the latent processes and propose two provable conditions under which temporally causal latent processes can be identified from their nonlinear mixtures. We propose LEAP, a theoretically-grounded framework that ext…
Mila - Québec AI Institute（米拉-魁北克人工智能研究所）；McGill University（麦吉尔大学）；ÉTS Montréal（蒙特利尔高等技术学院）
Learning predictors that do not rely on spurious correlations involves building causal representations. However, learning such a representation is very challenging. We, therefore, formulate the problem of learning a causal representation from high dimensional data and study causal recovery with synthetic data. This work introduces a latent variable decoder model, Decoder BCD, for Bayesian causal discovery and performs experiments in mildly supervised and unsupervised settings. We present a series of synthetic experiments to characterize important factors for causal discovery and show that usin…
Reconstructing the causal relationships behind the phenomena we observe is a fundamental challenge in all areas of science. Discovering causal relationships through experiments is often infeasible, unethical, or expensive in complex systems. However, increases in computational power allow us to process the ever-growing amount of data that modern science generates, leading to an emerging interest in the causal discovery problem from observational data. This work evaluates the LPCMCI algorithm, which aims to find generators compatible with a multi-dimensional, highly autocorrelated time series w…
Granger causality analysis, as one of the most popular time series causality methods, has been widely used in the economics, neuroscience. However, unobserved confounders is a fundamental problem in the observational studies, which is still not solved for the non-linear Granger causality. The application works often deal with this problem in virtue of the proxy variables, who can be treated as a measure of the confounder with noise. But the proxy variables has been proved to be unreliable, because of the bias it may induce. In this paper, we try to "recover" the unobserved confounders for the …
Identification of Causal Direction under an Arbitrary Number of Latent Confounders
School of Computer Science, Guangdong University of Technology（广东技术大学计算机科学学院）；Department of Philosophy, Carnegie Mellon University（卡内基梅隆大学哲学系）；College of Engineering, Shantou University（汕头大学工程学院）；Peng Cheng Laboratory, Shenzhen, China（深圳鹏城实验室）；Mohamed bin Zayed University of Artificial Intelligence（莫扎德大学人工智能大学）
Carnegie Mellon University（卡内基梅隆大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
One critical challenge of time-series modeling is how to learn and quickly correct the model under unknown distribution shifts. In this work, we propose a principled framework, called LiLY, to first recover time-delayed latent causal variables and identify their relations from measured temporal data under different distribution shifts. The correction step is then formulated as learning the low-dimensional change factors with a few samples from the new environment, leveraging the identified causal structure. Specifically, the framework factorizes unknown distribution shifts into transition dist…
Informatics Institute（信息学研究所）；University of Amsterdam（阿姆斯特丹大学）
We address the problem of causal discovery from data, making use of the recently proposed causal modeling framework of modular structural causal models (mSCM) to handle cycles, latent confounders and non-linearities. We introduce σ-connection graphs (σ-CG), a new class of mixed graphs (containing undirected, bidirected and directed edges) with additional structure, and extend the concept of σ-separation, the appropriate generalization of the well-known notion of d-separation in this setting, to apply to σ-CGs. We prove the closedness of σ-separation under marginalisation and conditioning and e…
Harvard School of Public Health（哈佛公共卫生学院）；University of Washington（华盛顿大学）
Probabilistic inference in graphical models is the task of computing marginal and conditional densities of interest from a factorized representation of a joint probability distribution. Inference algorithms such as variable elimination and belief propagation take advantage of constraints embedded in this factorization to compute such densities efficiently. In this paper, we propose an algorithm which computes interventional distributions in latent variable causal models represented by acyclic directed mixed graphs(ADMGs). To compute these distributions efficiently, we take advantage of a recur…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
How do we learn from biased data? Historical datasets often reflect historical prejudices; sensitive or protected attributes may affect the observed treatments and outcomes. Classification algorithms tasked with predicting outcomes accurately from these datasets tend to replicate these biases. We advocate a causal modeling approach to learning from biased data, exploring the relationship between fair classification and intervention. We propose a causal model in which the sensitive attribute confounds both the treatment and the outcome. Building on prior work in deep learning and generative mod…
Most causal discovery algorithms find causal structure among a set of observed variables. Learning the causal structure among latent variables remains an important open problem, particularly when using high-dimensional data. In this paper, we address a problem for which it is known that inputs cause outputs, and these causal relationships are encoded by a causal network among a set of an unknown number of latent variables. We developed a deep learning model, which we call a redundant input neural network (RINN), with a modified architecture and a regularized objective function to find causal r…
Differentiable Causal Discovery For Latent Hierarchical Causal Models
University of California San Diego（加利福尼亚大学圣迭戈分校）；Carnegie Mellon University（卡内基梅隆大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
University of Chicago（芝加哥大学）；Carnegie Mellon University（卡内基梅隆大学）
We study the problem of reconstructing a causal graphical model from data in the presence of latent variables. The main problem of interest is recovering the causal structure over the latent variables while allowing for general, potentially nonlinear dependence between the variables. In many practical problems, the dependence between raw observations (e.g. pixels in an image) is much less relevant than the dependence between certain high-level, latent features (e.g. concepts or objects), and this is the setting of interest. We provide conditions under which both the latent representations and …
Discovering Latent Causal Graphs from Spatiotemporal Data
Department of Computer Science and Engineering, University of California, San Diego, La Jolla, USA（计算机科学与工程系，加州大学圣地亚哥分校，拉乔拉，美国）；Halıcıoglu Data Science Institute, University of California, San Diego, La Jolla, USA（Halıcıoglu数据科学研究所，加州大学圣地亚哥分校，拉乔拉，美国）；Scripps Institution of Oceanography, University of California, San Diego, La Jolla, USA（斯克里普斯海洋研究所，加州大学圣地亚哥分校，拉乔拉，美国）
We present two online causal structure learning algorithms which can track changes in a causal structure and process data in a dynamic real-time manner. Standard causal structure learning algorithms assume that causal structure does not change during the data collection process, but in real-world scenarios, it does often change. Therefore, it is inappropriate to handle such changes with existing batch-learning approaches, and instead, a structure should be learned in an online manner. The online causal structure learning algorithms we present here can revise correlation values without reproces…
Using deep latent variable models in causal inference has attracted considerable interest recently, but an essential open question is their ability to yield consistent causal estimates. While they have demonstrated promising results and theory exists on some simple model formulations, we also know that causal effects are not even identifiable in general with latent variables. We investigate this gap between theory and empirical results with analytical considerations and extensive experiments under multiple synthetic and real-world data sets, using the causal effect variational autoencoder (CEV…
Identifiable Latent Polynomial Causal Models Through the Lens of Change
Australian Institute for Machine Learning, The University of Adelaide（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）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；Halicioğlu Data Science Institute (HDSI), University of California San Diego（加州大学圣地亚哥分校哈利西奥格鲁数据科学研究所）；Carnegie Mellon University（卡内基梅隆大学）
Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息科学与技术研究生院）；Graduate School of Arts and Sciences, The University of Tokyo（东京大学文理研究生院）
Identification of Nonlinear Latent Hierarchical Models
Carnegie Mellon University（卡内基梅隆大学）；University of California San Diego（加州大学圣迭戈分校）；Beijing Technology and Business University（北京工商大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
The tetrad constraint is a condition of which the satisfaction signals a rank reduction of a covariance submatrix and is used to design causal discovery algorithms that detects the existence of latent (unmeasured) variables, such as FOFC. Initially such algorithms only work for cases where the measured and latent variables are all Gaussian and have linear relations (Gaussian-Gaussian Case). It has been shown that a unidimentional latent variable model implies tetrad constraints when the measured and latent variables are all binary (Binary-Binary case). This paper proves that the tetrad constra…
A Recursive Decomposition Framework for Causal Structure Learning in the Presence of Latent Variables
Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China（深圳先进技术研究院，中国科学院，中国深圳）；College of Computer Science and Artificial Intelligence, Fudan University, Shanghai, China（复旦大学计算机科学与人工智能学院，中国上海）
Causal Effect Identification in LiNGAM Models with Latent Confounders
Technical University of Munich（慕尼黑工业大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Leiden University（莱顿大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）
Although understanding and characterizing causal effects have become essential in observational studies, it is challenging when the confounders are high-dimensional. In this article, we develop a general framework $\textit{CausalEGM}$ for estimating causal effects by encoding generative modeling, which can be applied in both binary and continuous treatment settings. Under the potential outcome framework with unconfoundedness, we establish a bidirectional transformation between the high-dimensional confounders space and a low-dimensional latent space where the density is known (e.g., multivaria…
Root causal analysis seeks to identify the set of initial perturbations that induce an unwanted outcome. In prior work, we defined sample-specific root causes of disease using exogenous error terms that predict a diagnosis in a structural equation model. We rigorously quantified predictivity using Shapley values. However, the associated algorithms for inferring root causes assume no latent confounding. We relax this assumption by permitting confounding among the predictors. We then introduce a corresponding procedure called Extract Errors with Latents (EEL) for recovering the error terms up to…
Latent variables may lead to spurious relationships that can be misinterpreted as causal relationships. In Bayesian Networks (BNs), this challenge is known as learning under causal insufficiency. Structure learning algorithms that assume causal insufficiency tend to reconstruct the ancestral graph of a BN, where bi-directed edges represent confounding and directed edges represent direct or ancestral relationships. This paper describes a hybrid structure learning algorithm, called CCHM, which combines the constraint-based part of cFCI with hill-climbing score-based learning. The score-based pro…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）；Massachusetts Institute of Technology（麻省理工学院）
Latent confounders---unobserved variables that influence both treatment and outcome---can bias estimates of causal effects. In some cases, these confounders are shared across observations, e.g. all students taking a course are influenced by the course's difficulty in addition to any educational interventions they receive individually. This paper shows how to semiparametrically model latent confounders that have this structure and thereby improve estimates of causal effects. The key innovations are a hierarchical Bayesian model, Gaussian processes with structured latent confounders (GP-SLC), an…
Informatics Institute（信息学研究所）；University of Amsterdam（阿姆斯特丹大学）
We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent confounders and arbitrary probability distributions. We also generalize adjustment criteria and formulas from the acyclic setting to the general one (i.e. ioSCMs). Such criteria then allow to estimate (conditional) causal effects from observational data that was (partially) gathered under selection bias and cycles. This generalizes the backdoor criterion, the se…
Centre d’Analyse et de Mathématique Sociales（社会分析与数学中心）；EHESS（高等社会科学研究院）；Centre de Mathématiques Appliquées（应用数学中心）；École Polytechnique（巴黎综合理工学院）；Inria XPOP（法国国家信息与自动化研究所XPOP团队）；Google Research（谷歌研究院）
Inferring causal effects of a treatment, intervention or policy from observational data is central to many applications. However, state-of-the-art methods for causal inference seldom consider the possibility that covariates have missing values, which is ubiquitous in many real-world analyses. Missing data greatly complicate causal inference procedures as they require an adapted unconfoundedness hypothesis which can be difficult to justify in practice. We circumvent this issue by considering latent confounders whose distribution is learned through variational autoencoders adapted to missing val…
We consider the task of learning a causal graph in the presence of latent confounders given i.i.d.~samples from the model. While current algorithms for causal structure discovery in the presence of latent confounders are constraint-based, we here propose a score-based approach. We prove that under assumptions weaker than faithfulness, any sparsest independence map (IMAP) of the distribution belongs to the Markov equivalence class of the true model. This motivates the \emph{Sparsest Poset} formulation - that posets can be mapped to minimal IMAPs of the true model such that the sparsest of these…
California Institute of Technology（加州理工学院）；UC Irvine（加州大学欧文分校）
Discovering the complete set of causal relations among a group of variables is a challenging unsupervised learning problem. Often, this challenge is compounded by the fact that there are latent or hidden confounders. When only observational data is available, the problem is ill-posed, i.e. the causal relationships are non-identifiable unless strong modeling assumptions are made. When interventions are available, we provide guarantees on identifiability and learnability under mild assumptions. We assume a linear structural equation model (SEM) with independent latent factors and directed acycli…
We consider recovering a causal graph in presence of latent variables, where we seek to minimize the cost of interventions used in the recovery process. We consider two intervention cost models: (1) a linear cost model where the cost of an intervention on a subset of variables has a linear form, and (2) an identity cost model where the cost of an intervention is the same, regardless of what variables it is on, i.e., the goal is just to minimize the number of interventions. Under the linear cost model, we give an algorithm to identify the ancestral relations of the underlying causal graph, achi…
Causal Discovery of Linear Non-Gaussian Causal Models with Unobserved Confounding
TUM School of Computation, Information and Technology（慕尼黑工业大学计算、信息与技术学院）；Technical University of Munich（慕尼黑工业大学）；University of British Columbia（不列颠哥伦比亚大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）
We present a sound and complete algorithm, called iterative causal discovery (ICD), for recovering causal graphs in the presence of latent confounders and selection bias. ICD relies on the causal Markov and faithfulness assumptions and recovers the equivalence class of the underlying causal graph. It starts with a complete graph, and consists of a single iterative stage that gradually refines this graph by identifying conditional independence (CI) between connected nodes. Independence and causal relations entailed after any iteration are correct, rendering ICD anytime. Essentially, we tie the …
