Structural Causal Models (SCMs) offer a unified probabilistic framework for observational, interventional and counterfactual reasoning. Existing approaches either convert an SCM into another tractable representation or rely on expressive generative models for approximate inference. Here we perform inference directly on the SCM via a message‑passing scheme that embeds both structural mechanisms and probability densities into a common finite‑dimensional function space. Forward propagation implements a numerical version of the Law of the Unconscious Statistician, and the propagated moments are turned back into densities by spline projection. The same representation yields a backward pass for conditioning, giving a single framework for causal inference. Using tensor‑product B‑spline bases, the algorithm reduces to numerical quadrature and linear algebra without explicitly forming conditional distributions. Experiments on randomly generated continuous SCMs show that the approximation matches Monte Carlo reference solutions and improves systematically as the basis resolution increases.
Review: The approach preserves inference accuracy while sidestepping explicit construction of high‑dimensional conditional densities, highlighting the promise of basis‑function approximations in causal inference.