Independence Test for Linear Non-Gaussian Data and Applications in Causal Discovery

Independence testing involves determining whether two variables are independent based on observed samples, which is a fundamental problem in causal discovery. Existing testing methods, such as HSIC, can theoretically detect broad forms of dependence, but may sacrifice statistical power when applied to limited samples with background knowledge of the distribution. In this paper, we focus on the linear non-Gaussian data, a widely supported model in scientific data analysis and causal discovery, where variables are linked linearly with noise terms that are non-Gaussian distributed. We provide a new theoretical characterization of independence in this case, showing that constancy of the conditional mean and variance is sufficient to guarantee independence under linear non-Gaussian models. Building on this result, we develop a kernel-based testing framework with provable asymptotic guarantees. Extensive experiments on synthetic and realworld datasets demonstrate that our method achieves higher power than existing approaches and significantly improves downstream causal discovery performance. Published as a conference paper at ICLR 2026 that leverage non-Gaussianity, we rely on tests that do not exploit this structure-akin to using a shotgun to shoot a butterfly, which is inefficient and potentially ineffective. A natural and pressing question arises: How can we design an independence test that is tailored to this well-established model class? By incorporating the model assumptions directly into the testing procedure, we can develop a method that is not only statistically more efficient but also conceptually simpler. This paper addresses this exact need. We propose a novel independence test specifically designed for the linear non-Gaussian data. We begin by providing a new characterization of independence in this setting. We show that, interestingly, for judging the independence of linear non-Gaussian data, it is enough to check the constancy of the conditional mean and the conditional variance. Based on this new characterization, we further designed a statistic that can test the conditions simultaneously. With derived asymptotic distributions, our method leverages the model constraints to achieve higher statistical power than generic alternatives, thereby providing a more robust foundation for causal discovery algorithms. We summarize our contributions as follows. • We propose a novel characterization of independence for linear mixtures of independent non-Gaussian components using only the conditional mean and conditional variance. • We propose a statistic and derive its corresponding asymptotic distributions to test independence. We also prove the equivalence of the statistic and the independence characterization. • We conduct extensive experiments on both synthetic and real-world data, which demonstrate the efficacy of our method. In addition, we integrate our testing method into an existing causal discovery algorithm and it outperforms other testing methods. BACKGROUND Problem Definition. For random variables X ∈ X and Y ∈ Y, where X and Y are their domains, we say X and Y are independent if P XY = P X P Y , denoted by X ⊥ ⊥ Y . Given a dataset D = (x i , y i ) n i=1 , where the pairs are independently and identically sampled from the joint distribution P XY , an independence test constructs a test statistic T based on D to test for hypotheses: The statistic T is then compared with a critical value to decide whether to reject the null hypothesis H 0 . The quality of an independence test is typically characterized by two quantities: the probability of incorrectly rejecting H 0 when it is true (Type I error), and the probability of failing to reject H 0 when it is false (Type II error). An ideal test maintains the Type I error at a user-specified significance level α, while achieving high statistical power (1-Type II error rate). A direct way to check for independence based on the definition. That is, estimate the probability densities of the joint distribution P XY and the marginal distribution P X , P Y , and then evaluate if P XY = P X P Y is satisfied almost surely. For example, mutual information measures the dependence strength between two variables using the KL divergence between P XY and P X P Y . However, estimating the probability densities from finite samples is difficult. Some distributions may even have no densities, which may further deteriorate the testing performance. Instead, (Jacod & Protter, 2004) provides an alternative characterization of the independence of random variables. Lemma 2.1. The random variables X and Y are independent if and only if Cov(f (X), g(Y )) = 0 for each pair (f, g) of bounded, continuous functions, i.e. f ∈ C b (X ) and g ∈ C b (Y). Lemma 2.1 provides a direct test criterion without the need for an intermediate density estimator. However, the space of bounded, continuous functions is too rich, which will raise the consistency issue. That is, the empirical estimate co