Who caused $Y$? Local identifiability for learning causal parents

We study identifiability and estimation of the direct causes, that is, the parents, of a designated target $Y$ in a structural causal model using only observational data. Our main assumption is an additive-noise model for $Y$: the value of $Y$ equals some function of its parents plus noise that is independent of those parents. We allow for general functional relationships and hidden confounding among all other variables. We prove that under a `no-backward' additive-noise condition the true parent set, $\mathrm{PA}(Y)$, is identified from the observational distribution by a simple population principle: among all candidate variable sets that make the regression residual of $Y$ independent of the regressors, choose the one with the smallest residual variance, and—if several tie—the smallest set. To justify the ``no-backward'' condition, we propose a novel identifiability scheme based on identifiability witnesses: we prove that in finite-dimensional analytic model classes, for which we have an identifiability witness (that is, a single identifiable model), residual independence for sets containing descendants of \(Y\) holds only on Lebesgue-null exceptional parameter sets. This scheme is strong enough to recover known identifiability results. For finite data, we propose Independent Risk Minimization (IndRM), which offers a simple, local, and non-interventional route to isolating the direct causes of $Y$ in multivariate systems, avoiding global model assumptions and full-graph search.