When Additive Noise Meets Unobserved Mediators: Bivariate Denoising Diffusion for Causal Discovery

Distinguishing cause and effect from bivariate observational data is a foundational problem in many disciplines, but challenging without additional assumptions. Additive noise models (ANMs) are widely used to enable sample-efficient bivariate causal discovery. However, conventional ANM-based methods fail when unobserved mediators corrupt the causal relationship between variables. This paper makes three key contributions: first, we rigorously characterize why standard ANM approaches break down in the presence of unmeasured mediators. Second, we demonstrate that prior solutions for hidden mediation are brittle in finite sample settings, limiting their practical utility. To address these gaps, we propose Bivariate Denoising Diffusion (BiDD) for causal discovery, a method designed to handle latent noise introduced by unmeasured mediators. Unlike prior methods that infer directionality through mean squared error loss comparisons, our approach introduces a novel independence test statistic: during the noising and denoising processes for each variable, we condition on the other variable as input and evaluate the independence of the predicted noise relative to this input. We prove asymptotic consistency of BiDD under the ANM, and conjecture that it performs well under hidden mediation. Experiments on synthetic and real-world data demonstrate consistent performance, outperforming existing methods in mediator-corrupted settings while maintaining strong performance in mediator-free settings.