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1. Identification of Nonlinear Latent Hierarchical Models

   Kong Lingjing, Huang Biwei ( July 2023 , arXivorg)

   Identifying latent variables and causal structures from observational data is essential to many real-world applications involving biological data, medical data, and unstructured data such as images and languages. However, this task can be highly challenging, especially when observed variables are generated by causally related latent variables and the relationships are nonlinear. In this work, we investigate the identification problem for nonlinear latent hierarchical causal models in which observed variables are generated by a set of causally related latent variables, and some latent variables may not have observed children. We show that the identifiability of both causal structure and latent variables can be achieved under mild assumptions: on causal structures, we allow for the existence of multiple paths between any pair of variables in the graph, which relaxes latent tree assumptions in prior work; on structural functions, we do not make parametric assumptions, thus permitting general nonlinearity and multi-dimensional continuous variables. Specifically, we first develop a basic identification criterion in the form of novel identifiability guarantees for an elementary latent variable model. Leveraging this criterion, we show that both causal structures and latent variables of the hierarchical model can be identified asymptotically by explicitly constructing an estimation procedure. To the best of our knowledge, our work is the first to establish identifiability guarantees for both causal structures and latent variables in nonlinear latent hierarchical models. 

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2. Identification of Nonlinear Latent Hierarchical Models

   Kong, Lingjing ; Huang, Biwei ; Xie, Feng ; Xing, Eric ; Chi, Yuejie ; Zhang, Kun ( December 2023 , Advances in neural information processing systems)

   Identifying latent variables and causal structures from observational data is essential to many real-world applications involving biological data, medical data, and unstructured data such as images and languages. However, this task can be highly challenging, especially when observed variables are generated by causally related latent variables and the relationships are nonlinear. In this work, we investigate the identification problem for nonlinear latent hierarchical causal models in which observed variables are generated by a set of causally related latent variables, and some latent variables may not have observed children. We show that the identifiability of causal structures and latent variables (up to invertible transformations) can be achieved under mild assumptions: on causal structures, we allow for multiple paths between any pair of variables in the graph, which relaxes latent tree assumptions in prior work; on structural functions, we permit general nonlinearity and multi-dimensional continuous variables, alleviating existing work's parametric assumptions. Specifically, we first develop an identification criterion in the form of novel identifiability guarantees for an elementary latent variable model. Leveraging this criterion, we show that both causal structures and latent variables of the hierarchical model can be identified asymptotically by explicitly constructing an estimation procedure. To the best of our knowledge, our work is the first to establish identifiability guarantees for both causal structures and latent variables in nonlinear latent hierarchical models. 

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3. Triad Constraints for Learning Causal Structure of Latent Variables

   Ruichu Cai, Feng Xie ( December 2019 , Advances in neural information processing systems)

   Learning causal structure from observational data has attracted much attention,and it is notoriously challenging to find the underlying structure in the presenceof confounders (hidden direct common causes of two variables). In this paper,by properly leveraging the non-Gaussianity of the data, we propose to estimatethe structure over latent variables with the so-called Triad constraints: we design a form of "pseudo-residual" from three variables, and show that when causal relations are linear and noise terms are non-Gaussian, the causal direction between the latent variables for the three observed variables is identifiable by checking a certain kind of independence relationship. In other words, the Triad constraints help us to locate latent confounders and determine the causal direction between them. This goes far beyond the Tetrad constraints and reveals more information about the underlying structure from non-Gaussian data. Finally, based on the Triad constraints, we develop a two-step algorithm to learn the causal structure corresponding to measurement models. Experimental results on both synthetic and real data demonstrate the effectiveness and reliability of our method. 

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4. Causal Inference with Latent Variables: Recent Advances and Future Prospectives

   https://doi.org/10.1145/3637528.3671450  

   Zhu, Yaochen ; He, Yinhan ; Ma, Jing ; Hu, Mengxuan ; Li, Sheng ; Li, Jundong ( August 2024 , Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining)

   Causality lays the foundation for the trajectory of our world. Causal inference (CI), which aims to infer intrinsic causal relations among variables of interest, has emerged as a crucial research topic. Nevertheless, the lack of observation of important variables (e.g., confounders, mediators, exogenous variables, etc.) severely compromises the reliability of CI methods. The issue may arise from the inherent difficulty in measuring the variables. Additionally, in observational studies where variables are passively recorded, certain covariates might be inadvertently omitted by the experimenter. Depending on the type of unobserved variables and the specific CI task, various consequences can be incurred if these latent variables are carelessly handled, such as biased estimation of causal effects, incomplete understanding of causal mechanisms, lack of individual-level causal consideration, etc. In this survey, we provide a comprehensive review of recent developments in CI with latent variables. We start by discussing traditional CI techniques when variables of interest are assumed to be fully observed. Afterward, under the taxonomy of circumvention and inference-based methods, we provide an in-depth discussion of various CI strategies to handle latent variables, covering the tasks of causal effect estimation, mediation analysis, counterfactual reasoning, and causal discovery. Furthermore, we generalize the discussion to graph data where interference among units may exist. Finally, we offer fresh aspects for further advancement of CI with latent variables, especially new opportunities in the era of large language models (LLMs). 

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5. Causal Discovery and Forecasting in Nonstationary Environments with State-Space Models

   Biwei Huang, Kun Zhang ( January 2019 , Proceedings of the 36th International Conference on Machine Learning)

   In many scientific fields, such as economics and neuroscience, we are often faced with nonstationary time series, and concerned with both finding causal relations and forecasting the values of variables of interest, both of which are particularly challenging in such nonstationary environments. In this paper, we study causal discovery and forecasting for nonstationary time series. By exploiting a particular type of state-space model to represent the processes, we show that nonstationarity helps to identify the causal structure, and that forecasting naturally benefits from learned causal knowledge. Specifically, we allow changes in both causal strengths and noise variances in the nonlinear state-space models, which, interestingly, renders both the causal structure and model parameters identifiable. Given the causal model, we treat forecasting as a problem in Bayesian inference in the causal model, which exploits the time-varying property of the data and adapts to new observations in a principled manner. Experimental results on synthetic and real-world data sets demonstrate the efficacy of the proposed methods. 

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