In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply. We first extend the half-trek criterion to cover a broader class of models and apply our extension to finding testable constraints implied by the model. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification and constraint discovery power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification and constraint discovery algorithms for non-parametric models.

In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply.

The observational characteristics of a linear structural equation model can be effectively described by polynomial constraints on the observed covariance matrix. However, these polynomials can be exponentially large, making them impractical for many purposes. In this paper, we present a graphical notation for many of these polynomial constraints. The expressive power of this notation is investigated both theoretically and empirically.

In this paper we propose a new method to learn the underlying acyclic mixed graph of a linear non-Gaussian structural equation model given observational data. We build on an algorithm proposed by Wang and Drton, and we show that one can augment the hidden variable structure of the recovered model by learning {\em multidirected edges} rather than only directed and bidirected ones. Multidirected edges appear when more than two of the observed variables have a hidden common cause. We detect the presence of such hidden causes by looking at higher order cumulants and exploiting the multi-trek rule. Our method recovers the correct structure when the underlying graph is a bow-free acyclic mixed graph with potential multi-directed edges.

In this work, we consider the problem of robust parameter estimation from observational data in the context of linear structural equation models (LSEMs). LSEMs are a popular and well-studied class of models for inferring causality in the natural and social sciences. One of the main problems related to LSEMs is to recover the model parameters from the observational data. Under various conditions on LSEMs and the model parameters the prior work provides efficient algorithms to recover the parameters. However, these results are often about generic identifiability. In practice, generic identifiability is not sufficient and we need robust identifiability: small changes in the observational data should not affect the parameters by a huge amount. Robust identifiability has received far less attention and remains poorly understood. Sankararaman et al. (2019) recently provided a set of sufficient conditions on parameters under which robust identifiability is feasible. However, a limitation of their work is that their results only apply to a small sub-class of LSEMs, called ``bow-free paths.'' In this work, we significantly extend their work along multiple dimensions. First, for a large and well-studied class of LSEMs, namely ``bow free'' models, we provide a sufficient condition on model parameters under which robust identifiability holds, thereby removing the restriction of paths required by prior work. We then show that this sufficient condition holds with high probability which implies that for a large set of parameters robust identifiability holds and that for such parameters, existing algorithms already achieve robust identifiability. Finally, we validate our results on both simulated and real-world datasets.

In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply. We first extend the half-trek criterion to cover a broader class of models and apply our extension to finding testable constraints implied by the model. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification and constraint discovery power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification and constraint discovery algorithms for non-parametric models. Papers published at the Neural Information Processing Systems Conference.

We consider the numerical stability of the parameter recovery problem in Linear Structural Equation Model ($\LSEM$) of causal inference. A long line of work starting from Wright (1920) has focused on understanding which sub-classes of $\LSEM$ allow for efficient parameter recovery. Despite decades of study, this question is not yet fully resolved. The goal of this paper is complementary to this line of work; we want to understand the stability of the recovery problem in the cases when efficient recovery is possible. Numerical stability of Pearl's notion of causality was first studied in Schulman and Srivastava (2016) using the concept of condition number where they provide ill-conditioned examples. In this work, we provide a condition number analysis for the $\LSEM$. First we prove that under a sufficient condition, for a certain sub-class of $\LSEM$ that are \emph{bow-free} (Brito and Pearl (2002)), the parameter recovery is stable. We further prove that \emph{randomly} chosen input parameters for this family satisfy the condition with a substantial probability. Hence for this family, on a large subset of parameter space, recovery is numerically stable. Next we construct an example of $\LSEM$ on four vertices with \emph{unbounded} condition number. We then corroborate our theoretical findings via simulations as well as real-world experiments for a sociology application. Finally, we provide a general heuristic for estimating the condition number of any $\LSEM$ instance.

Despite their popularity, many questions about the algebraic constraints imposed by linear structural equation models remain open problems. For causal discovery, two of these problems are especially important: the enumeration of the constraints imposed by a model, and deciding whether two graphs define the same statistical model. We show how the half-trek criterion can be used to make progress in both of these problems. We apply our theoretical results to a small-scale model selection problem, and find that taking the additional algebraic constraints into account may lead to significant improvements in model selection accuracy.

In this paper, we address the problems of identifying linear structural equation models and discovering the constraints they imply. We first extend the half-trek criterion to cover a broader class of models and apply our extension to finding testable constraints implied by the model. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification and constraint discovery power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification and constraint discovery algorithms for non-parametric models.

In this paper, we address the problem of identifying linear structural equation models. We first extend the edge set half-trek criterion to cover a broader class of models. We then show that any semi-Markovian linear model can be recursively decomposed into simpler sub-models, resulting in improved identification power. Finally, we show that, unlike the existing methods developed for linear models, the resulting method subsumes the identification algorithm of non-parametric models.