We study the problem of learning the causal structure of a latent tree model only from the observational data. Prior works often assume either a sufficient number of measured variables or the observed variables can only be the child of latent variables (known as measurement assumption). However, they may yield incorrect or uninformative results when some observed variables also cause the latent variable, or when the number of measured variables is less than two. In this paper, we focus on the linear non-Gaussian latent polytree model, where the observed and latent variables can exhibit arbitrary causal dependence and the number of child variables for each latent variable may be only one. By leveraging the non-Gaussianity within the causal model, we introduce rank constraints of high-order cumulants. These constraints align with trek separation within the causal graph and enable the identification of exogenous variables for the relative set. Such properties have intriguing possibilities for identifying the entire latent polytree structure, including not only the number of latent variables but also causal directions. Consequently, we develop an identification algorithm to learn latent polytree by only using the rank constraints of high-order cumulants, and we verify its effectiveness in simulation experiments.

Networked interference, also known as the peer effect in social science and spillover effect in economics, has drawn increasing interest across various domains. This phenomenon arises when a unit’s treatment and outcome are influenced by the actions of its peers, posing significant challenges to causal inference, particularly in treatment assignment and effect estimation in real applications, due to the violation of the SUTVA assumption. While extensive graph models have been developed to identify treatment effects, these models often rely on structural assumptions about networked interference, assuming it to be identical to the social network, which can lead to misspecification issues in real applications. To address these challenges, we propose an Interference-Agnostic Causal Graph Transformer (CauGramer), which aggregates peers information via $L$-order Graph Transformer and employs cross-attention to infer aggregation function for learning interference representations. By integrating confounder balancing and minimax moment constraints, CauGramer fully incorporates peer information, enabling robust treatment effect estimation. Extensive experiments on two widely-used benchmarks demonstrate the effectiveness and superiority of CauGramer. The code is available at https://github.com/anpwu/CauGramer.

We study the problem of learning discrete latent variable causal structures from mixed-type observational data. Traditional methods, such as those based on the tensor rank condition, are designed to identify discrete latent structure models and provide robust identification bounds for discrete causal models. However, when observed variables—specifically, those representing the children of latent variables—are collected at various levels with continuous data types, the tensor rank condition is not applicable, limiting further causal structure learning for latent variables. In this paper, we consider a more general case where observed variables can be either continuous or discrete, and further allow for scenarios where multiple latent parents cause the same set of observed variables. We show that, under the completeness condition, it is possible to discretize the data in a way that satisfies the full-rank assumption required by the tensor rank condition. This enables the identifiability of discrete latent structure models within mixed-type observational data. Moreover, we introduce the two-sufficient measurement condition, a more general structural assumption under which the tensor rank condition holds and the underlying latent causal structure is identifiable by a proposed two-stage identification algorithm. Extensive experiments on both simulated and real-world data validate the effectiveness of our method.

Discovering dependence patterns between variables from observational data is a fundamental issue in data analysis. However, existing testing methods often fail to detect subtle yet critical patterns that occur within small regions of the data distribution--patterns we term rare dependence. These rare dependencies obscure the true underlying dependence structure in variables, particularly in causal discovery tasks. To address this issue, we propose a novel testing method that combines kernel-based (conditional) independence testing with adaptive sample importance reweighting. By learning and assigning higher importance weights to data points exhibiting significant dependence, our method amplifies the patterns and can detect them successfully. Theoretically, we analyze the asymptotic distributions of the statistics in this method and show the uniform bound of the learning scheme. Furthermore, we integrate our tests into the PC algorithm, a constraint-based approach for causal discovery, equipping it to uncover causal relationships even in the presence of rare dependence. Empirical evaluation of synthetic and real-world datasets comprehensively demonstrates the efficacy of our method.

Testing conditional independence is a critical task, particularly in causal discovery and learning in Bayesian networks. However, in many real-world scenarios, variables are often measured with errors, such as those introduced by insufficient measurement accuracy, complicating the testing process. This paper focuses on testing conditional independence in the linear non-Gaussian measurement error model, under the condition that measurement error noise follows a Gaussian distribution. By leveraging high-order cumulants, we derive rank constraints on the cumulant matrix and establish their role in effectively assessing conditional independence, even in the presence of measurement errors. Based on these theoretical results, we leverage the rank constraints of the cumulant matrix as a tool for conditional independence testing and incorporate it into the PC algorithm, resulting in the PC-ME algorithm — a method designed to learn causal structures from observed data while accounting for measurement errors. Experimental results demonstrate that the proposed method outperforms existing approaches, particularly in cases other methods encounter difficulties.

Time series imputation is one of the most challenging problems and has broad applications in various fields like health care and the Internet of Things. Existing methods mainly aim to model the temporally latent dependencies and the generation process from the observed time series data. In real-world scenarios, different types of missing mechanisms, like MAR (Missing At Random) and MNAR (Missing Not At Random), can occur in time series data. However, existing methods often overlook the difference among the aforementioned missing mechanisms and use a single model for time series imputation, which can easily lead to misleading results due to mechanism mismatching. In this paper, we propose a framework for the time series imputation problem by exploring Different Missing Mechanisms (DMM in short) and tailoring solutions accordingly. Specifically, we first analyze the data generation processes with temporal latent states and missing cause variables for different mechanisms. Sequentially, we model these generation processes via variational inference and estimate prior distributions of latent variables via a normalizing flow-based neural architecture. Furthermore, we establish identifiability results under the nonlinear independent component analysis framework to show that latent variables are identifiable. Experimental results show that our method surpasses existing time series imputation techniques across various datasets with different missing mechanisms, demonstrating its effectiveness in real-world applications.

Conditional independence (CI) testing is an important problem, especially in causal discovery. Most testing methods assume that all variables are fully observable and then test the CI among the observed data. Such an assumption is often untenable beyond applications dealing with, e.g., psychological analysis about the mental health status and medical diagnosing (researchers need to consider the existence of latent variables in these scenarios); and typically adopted latent CI test schemes mainly suffer from robust or efficient issues. Accordingly, this article investigates the problem of testing CI between latent variables. To this end, we offer an auxiliary regression-based CI (AReCI) test by taking the measured variable as the surrogate variable of the latent variables to conduct the regression over the latent variables under the linear causal models, in which each latent variable has some certain measured variables. Specifically, given a pair of latent variables$L_X$and$L_Y$, and a corresponding latent variable set$\mathcal{L}_{O}$,$L_X \CI L_Y | \mathcal{L}_{O}$holds if and only if$A_{\{L_X\}}-\omega_1^\intercal A^{\prime}_{\{\mathcal{L}_{O}\}}$and$A_{\{L_Y\}}-\omega_2^\intercal A^{\prime\prime}_{\{\mathcal{L}_{O}\}}$are statistically independent, where$A^{\prime}$and$A^{\prime\prime}$are the two disjoint subset of the measured variable for the corresponding latent variables,$A^{\prime}_{\{\mathcal{L}_{O}\}} \cap A^{\prime\prime}_{\{\mathcal{L}_{O}\}} =\emptyset$, and$\omega_1$is a parameter vector characterized from the cross covariance between$A_{\{L_X\}}$and$A^{\prime}_{\{\mathcal{L}_{O}\}}$, and$\omega_{2}$is a parameter vector characterized from the cross covariance between$A_{\{L_Y\}}$and$A^{\prime\prime}_{\{\mathcal{L}_{O}\}}$. We theoretically show that the AReCI test is capable of addressing both Gaussian and non-Gaussian data. In addition, we find that the well-known partial correlation test can be seen as a special case of the AReCI test. Finally, we devise a causal discovery method by using the AReCI test as the CI test. The experimental results on synthetic and real-world data illustrate the effectiveness of our method.

Missing data are an unavoidable complication frequently encountered in many causal discovery tasks. When a missing process depends on the missing values themselves (known as self-masking missingness), the recovery of the joint distribution becomes unattainable, and detecting the presence of such self-masking missingness remains a perplexing challenge. Consequently, due to the inability to reconstruct the original distribution and to discern the underlying missingness mechanism, simply applying existing causal discovery methods would lead to wrong conclusions. In this work, we found that the recent advances additive noise model has the potential for learning causal structure under the existence of the self-masking missingness. With this observation, we aim to investigate the identification problem of learning causal structure from missing data under an additive noise model with different missingness mechanisms, where the `no self-masking missingness' assumption can be eliminated appropriately. Specifically, we first elegantly extend the scope of identifiability of causal skeleton to the case with weak self-masking missingness (i.e., no other variable could be the cause of self-masking indicators except itself). We further provide the sufficient and necessary identification conditions of the causal direction under additive noise model and show that the causal structure can be identified up to an IN-equivalent pattern. We finally propose a practical algorithm based on the above theoretical results on learning the causal skeleton and causal direction. Extensive experiments on synthetic and real data demonstrate the efficiency and effectiveness of the proposed algorithms.

Count data naturally arise in many fields, such as finance, neuroscience, and epidemiology, and discovering causal structure among count data is a crucial task in various scientific and industrial scenarios. One of the most common characteristics of count data is the inherent branching structure described by a binomial thinning operator and an independent Poisson distribution that captures both branching and noise. For instance, in a population count scenario, mortality and immigration contribute to the count, where survival follows a Bernoulli distribution, and immigration follows a Poisson distribution. However, causal discovery from such data is challenging due to the non-identifiability issue: a single causal pair is Markov equivalent, i.e., X->Y and Y->X are distributed equivalent. Fortunately, in this work, we found that the causal order from X to its child Y is identifiable if X is a root vertex and has at least two directed paths to Y, or the ancestor of X with the most directed path to X has a directed path to Y without passing X. Specifically, we propose a Poisson Branching Structure Causal Model (PB-SCM) and perform a path analysis on PB-SCM using high-order cumulants. Theoretical results establish the connection between the path and cumulant and demonstrate that the path information can be obtained from the cumulant. With the path information, causal order is identifiable under some graphical conditions. A practical algorithm for learning causal structure under PB-SCM is proposed and the experiments demonstrate and verify the effectiveness of the proposed method.

Conventional causal discovery approaches, which seek to uncover causal relationships among measured variables, are typically fragile to the presence of latent variables. While various methods have been developed to address this confounding issue, they often rely on strong assumptions about the underlying causal structure. In this paper, we consider a general scenario where measured and latent variables collectively form a partially observed causally sufficient linear system and latent variables may be anywhere in the causal structure. We theoretically show that with the aid of high-order statistics, the causal graph is (almost) fully identifiable if, roughly speaking, each latent set has a sufficient number of pure children, which can be either latent or measured. Naturally, LiNGAM, a model without latent variables, is encompassed as a special case. Based on the identification theorem, we develop a principled algorithm to identify the causal graph by testing for statistical independence involving only measured variables in specific manners. Experimental results show that our method effectively recovers the causal structure, even when latent variables are influenced by measured variables.

Recently, interest has grown in the use of proxy variables of unobserved confounding for inferring the causal effect in the presence of unmeasured confounders from observational data. One difficulty inhibiting the practical use is finding valid proxy variables of unobserved confounding to a target causal effect of interest. These proxy variables are typically justified by background knowledge. In this paper, we investigate the estimation of causal effects among multiple treatments and a single outcome, all of which are affected by unmeasured confounders, within a linear causal model, without prior knowledge of the validity of proxy variables. To be more specific, we first extend the existing proxy variable estimator, originally addressing a single unmeasured confounder, to accommodate scenarios where multiple unmeasured confounders exist between the treatments and the outcome. Subsequently, we present two different sets of precise identifiability conditions for selecting valid proxy variables of unmeasured confounders, based on the second-order statistics and higher-order statistics of the data, respectively. Moreover, we propose two data-driven methods for the selection of proxy variables and for the unbiased estimation of causal effects. Theoretical analysis demonstrates the correctness of our proposed algorithms. Experimental results on both synthetic and real-world data show the effectiveness of the proposed approach.

Unobserved discrete data are ubiquitous in many scientific disciplines, and how to learn the causal structure of these latent variables is crucial for uncovering data patterns. Most studies focus on the linear latent variable model or impose strict constraints on latent structures, which fail to address cases in discrete data involving non-linear relationships or complex latent structures. To achieve this, we explore a tensor rank condition on contingency tables for an observed variable set $\mathbf{X}_p$, showing that the rank is determined by the minimum support of a specific conditional set (not necessary in $\mathbf{X}_p$) that d-separates all variables in $\mathbf{X}_p$. By this, one can locate the latent variable through probing the rank on different observed variables set, and further identify the latent causal structure under some structure assumptions. We present the corresponding identification algorithm and conduct simulated experiments to verify the effectiveness of our method. In general, our results elegantly extend the identification boundary for causal discovery with discrete latent variables and expand the application scope of causal discovery with latent variables.

We investigate the challenging task of learning causal structure in the presence of latent variables, including locating latent variables, determining their quantity, and identifying causal relationships among both latent and observed variables. To address this, we propose a Generalized Independent Noise (GIN) condition for linear non-Gaussian acyclic causal models that incorporate latent variables, which establishes the independence between a linear combination of certain measured variables and some other measured variables. Specifically, for two observed random vectors $\bf{Y}$ and $\bf{Z}$, GIN holds if and only if $\omega^{\intercal}\mathbf{Y}$ and $\mathbf{Z}$ are statistically independent, where $\omega$ is a non-zero parameter vector determined by the cross-covariance between $\mathbf{Y}$ and $\mathbf{Z}$. We then give necessary and sufficient graphical criteria of the GIN condition in linear non-Gaussian acyclic causal models. From a graphical perspective, roughly speaking, GIN implies the existence of a set $\mathcal{S}$ such that $\mathcal{S}$ is causally earlier (w.r.t. the causal ordering) than $\mathbf{Y}$, and that every active (collider-free) path between $\mathbf{Y}$ and $\mathbf{Z}$ must contain a node from $\mathcal{S}$. Interestingly, we find that the independent noise condition (i.e., if there is no confounder, causes are independent of the residual derived from regressing the effect on the causes) can be seen as a special case of GIN. With such a connection between GIN and latent causal structures, we further leverage the proposed GIN condition, together with a well-designed search procedure, to efficiently estimate Linear, Non-Gaussian Latent Hierarchical Models (LiNGLaHs), where latent confounders may also be causally related and may even follow a hierarchical structure. We show that the underlying causal structure of a LiNGLaH is identifiable in light of GIN conditions under mild assumptions. Experimental results on both synthetic and three real-world data sets show the effectiveness of the proposed approach.

We study the problem of learning hierarchical causal structure among latent variables from measured variables. While some existing methods are able to recover the latent hierarchical causal structure, they mostly suffer from restricted assumptions, including the tree-structured graph constraint, no ``triangle" structure, and non-Gaussian assumptions. In this paper, we relax these restrictions above and consider a more general and challenging scenario where the beyond tree-structured graph, the ``triangle" structure, and the arbitrary noise distribution are allowed. We investigate the identifiability of the latent hierarchical causal structure and show that by using second-order statistics, the latent hierarchical structure can be identified up to the Markov equivalence classes over latent variables. Moreover, some directions in the Markov equivalence classes of latent variables can be further identified using partially non-Gaussian data. Based on the theoretical results above, we design an effective algorithm for learning the latent hierarchical causal structure. The experimental results on synthetic data verify the effectiveness of the proposed method.

An important problem across multiple disciplines is to infer and understand meaningful latent variables. One strategy commonly used is to model the measured variables in terms of the latent variables under suitable assumptions on the connectivity from the latents to the measured (known as measurement model). Furthermore, it might be even more interesting to discover the causal relations among the latent variables (known as structural model). Recently, some methods have been proposed to estimate the structural model by assuming that the noise terms in the measured and latent variables are non-Gaussian. However, they are not suitable when some of the noise terms become Gaussian. To bridge this gap, we investigate the problem of identification of the structural model with arbitrary noise distributions. We provide necessary and sufficient condition under which the structural model is identifiable: it is identifiable iff for each pair of adjacent latent variables Lx, Ly, (1) at least one of Lx and Ly has non-Gaussian noise, or (2) at least one of them has a non-Gaussian ancestor and is not d-separated from the non-Gaussian component of this ancestor by the common causes of Lx and Ly. This identifiability result relaxes the non-Gaussianity requirements to only a (hopefully small) subset of variables, and accordingly elegantly extends the application scope of the structural model. Based on the above identifiability result, we further propose a practical algorithm to learn the structural model. We verify the correctness of the identifiability result and the effectiveness of the proposed method through empirical studies.

Traditional causal discovery methods mainly focus on estimating causal relations among measured variables, but in many real-world problems, such as questionnaire-based psychometric studies, measured variables are generated by latent variables that are causally related. Accordingly, this paper investigates the problem of discovering the hidden causal variables and estimating the causal structure, including both the causal relations among latent variables and those between latent and measured variables. We relax the frequently-used measurement assumption and allow the children of latent variables to be latent as well, and hence deal with a specific type of latent hierarchical causal structure. In particular, we define a minimal latent hierarchical structure and show that for linear non-Gaussian models with the minimal latent hierarchical structure, the whole structure is identifiable from only the measured variables. Moreover, we develop a principled method to identify the structure by testing for Generalized Independent Noise (GIN) conditions in specific ways. Experimental results on both synthetic and real-world data show the effectiveness of the proposed approach.