Question: What are the innovation points of this work？
Evidence: Understanding causal structures of systems is a fundamental question in science and has been studied in various fields such as biology, economics, and social science. While conducting randomized experiments is the most effective way to identify causal structures, it is often impossible for ethical, technical or cost reasons. Therefore, it is important to develop causal discovery methods that infer causal structures from uncontrolled observational data.
Using structural equation models (SEMs) is one major approach in causal discovery. In SEMbased methods, causal structures are assumed to follow parameterized causal models and inferred by estimating the parameters from data. An important property of causal SEMs is identifiability that is a theoretical guarantee that true causal structures can be uniquely identifiable if the joint distributions of data are known. Constructing flexible identifiable models and then developing their estimation methods are the main focuses in SEM-based study.
Various identifiable SEMs and their estimation methods have been proposed so far. The linear non-Gaussian acyclic model (LiNGAM) is one of the most studied models. In LiNGAM, a causal structure is represented as a directed acyclic graph (DAG) and each effect variable is generated as the linear combination of cause variables followed by an additive unobserved noise that is independent of the causes. There has been proposed an efficient estimation method named DirectLiNGAM. While LiNGAM is a simple and well-studied model, it may suffer from degradation in complex real-world applications due to the linear assumption. The additive noise model (ANM) is proposed as a more flexible model that can handle nonlinear causal relations. In ANM, cause variables are transformed nonlinearly and then added to an unobserved noise independent from the causes. ANM is proved identifiable except for some trivial cases and an estimation method based on regressions, named regression with subsequent independence test (RESIT), is proposed. As a more realistic model, proposed the post-nonlinear (PNL) model inspired by real-world data observation processes. In the PNL model, cause variables are transformed and added to a noise variable in the same manner as ANM and then transformed again by a nonlinear invertible function. The second transformation represents the sensor or measurement distortion, which is frequently encountered in practice. Although the PNL model is one of the most flexible identifiable causal SEMs, which covers ANM and LiNGAM, its estimation methods are not well-developed except the bivariate case. proposed an estimation method of the bivariate PNL model and a causal discovery procedure based on it, named autoencoder-based causal discovery for the PNL model (AbPNL). AbPNL represents the two nonlinearities with neural networks and estimates them by minimizing two loss terms corresponding to two assumptions of the model: the independence of the cause and the noise and the invertibility of the second nonlinearity. The losses are defined as Hilbert-Schmidt independence criterion (HSIC) and a reconstruction loss, respectively, and minimized directly by a stochastic gradient descent method. Given samples of two variables, AbPNL estimates the models of the two candidate causal directions and infer the true direction as the one that fits the model better. While AbPNL is easy to use owing to no assumptions on neural network structures and the straightforward optimization procedure, it is limited to the bivariate cases and cannot be applied to multivariate cases unless an exhaustive search is performed, which is usually infeasible in practice.
In this paper, we propose a new method for multivariate nonlinear causal discovery based on the PNL model. First, we extend the estimation of the bivariate PNL model of AbPNL to the multi-cause model with multiple cause variables and a single effect variable. Then, we iterate the multi-cause model estimation to identify the causal order of the variables in the same manner as the ANM case in RESIT. Finally, we prune redundant edges of the causal DAG constructed from the causal order to obtain the true causal structure. We also show the theoretical validity of the proposed procedure in the multivariate PNL model. We apply the proposed method to synthetic and real-world datasets and show its effectiveness.
In this paper, we propose a new method for multivariate nonlinear causal discovery based on the
PNL model. First, we extend the estimation of the bivariate PNL model of AbPNL to the multi-cause
model with multiple cause variables and a single effect variable. Then, we iterate the multi-cause
model estimation to identify the causal order of the variables in the same manner as the ANM case
in RESIT. Finally, we prune redundant edges of the causal DAG constructed from the causal order
to obtain the true causal structure. We also show the theoretical validity of the proposed procedure
in the multivariate PNL model. We apply the proposed method to synthetic and real-world datasets
and show its effectiveness.
Answer: In this paper, we aim to develop a new multivariate causal discovery method based on the PNL model. Despite its theoretical foundations, the practical estimation method for the multivariate PNL model is not well developed. We believe that our proposal increases the applicability of SEM-based causal discovery methods and promotes their use in real-world problems.