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1. Order-based structure learning without score equivalence

   https://doi.org/10.1093/biomet/asad052  

   Chang, Hyunwoong; Cai, James J; Zhou, Quan (September 2023, Biometrika)

   We propose an empirical Bayes formulation of the structure learning problem, where the prior specification assumes that all node variables have the same error variance, an assumption known to ensure the identifiability of the underlying causal directed acyclic graph. To facilitate efficient posterior computation, we approximate the posterior probability of each ordering by that of a best directed acyclic graph model, which naturally leads to an order-based Markov chain Monte Carlo algorithm. Strong selection consistency for our model in high-dimensional settings is proved under a condition that allows heterogeneous error variances, and the mixing behaviour of our sampler is theoretically investigated. Furthermore, we propose a new iterative top-down algorithm, which quickly yields an approximate solution to the structure learning problem and can be used to initialize the Markov chain Monte Carlo sampler. We demonstrate that our method outperforms other state-of-the-art algorithms under various simulation settings, and conclude the paper with a single-cell real-data study illustrating practical advantages of the proposed method. 

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2. 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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3. Causal Discovery in Linear Latent Variable Models Subject to Measurement Error

   Yuqin Yang, AmirEmad Ghassami (January 2022, Advances in Neural Information Processing Systems 35 (NeurIPS 2022))

   S. Koyejo; S. Mohamed; A. Agarwal; D. Belgrave; K. Cho, A. Oh (Ed.)

   We focus on causal discovery in the presence of measurement error in linear systems where the mixing matrix, i.e., the matrix indicating the independent exogenous noise terms pertaining to the observed variables, is identified up to permutation and scaling of the columns. We demonstrate a somewhat surprising connection between this problem and causal discovery in the presence of unobserved parentless causes, in the sense that there is a mapping, given by the mixing matrix, between the underlying models to be inferred in these problems. Consequently, any identifiability result based on the mixing matrix for one model translates to an identifiability result for the other model. We characterize to what extent the causal models can be identified under a two-part faithfulness assumption. Under only the first part of the assumption (corresponding to the conventional definition of faithfulness), the structure can be learned up to the causal ordering among an ordered grouping of the variables but not all the edges across the groups can be identified. We further show that if both parts of the faithfulness assumption are imposed, the structure can be learned up to a more refined ordered grouping. As a result of this refinement, for the latent variable model with unobserved parentless causes, the structure can be identified. Based on our theoretical results, we propose causal structure learning methods for both models, and evaluate their performance on synthetic data. 

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4. A polynomial-time algorithm for learning nonparametric causal graphs

   Gao, Ming; Ding, Yi; Aragam, Bryon (October 2020, Advances in Neural Information Processing Systems 33 (NeurIPS 2020))

   null (Ed.)

   We establish finite-sample guarantees for a polynomial-time algorithm for learning a nonlinear, nonparametric directed acyclic graphical (DAG) model from data. The analysis is model-free and does not assume linearity, additivity, independent noise, or faithfulness. Instead, we impose a condition on the residual variances that is closely related to previous work on linear models with equal variances. Compared to an optimal algorithm with oracle knowledge of the variable ordering, the additional cost of the algorithm is linear in the dimension d and the number of samples n. Finally, we compare the proposed algorithm to existing approaches in a simulation study. 

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5. The DeCAMFounder: nonlinear causal discovery in the presence of hidden variables

   https://doi.org/10.1093/jrsssb/qkad071  

   Agrawal, Raj; Squires, Chandler; Prasad, Neha; Uhler, Caroline (July 2023, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Many real-world decision-making tasks require learning causal relationships between a set of variables. Traditional causal discovery methods, however, require that all variables are observed, which is often not feasible in practical scenarios. Without additional assumptions about the unobserved variables, it is not possible to recover any causal relationships from observational data. Fortunately, in many applied settings, additional structure among the confounders can be expected. In particular, pervasive confounding is commonly encountered and has been utilised for consistent causal estimation in linear causal models. In this article, we present a provably consistent method to estimate causal relationships in the nonlinear, pervasive confounding setting. The core of our procedure relies on the ability to estimate the confounding variation through a simple spectral decomposition of the observed data matrix. We derive a DAG score function based on this insight, prove its consistency in recovering a correct ordering of the DAG, and empirically compare it to previous approaches. We demonstrate improved performance on both simulated and real datasets by explicitly accounting for both confounders and nonlinear effects. 

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