Beyond Additivity: Causal Discovery in Location-Scale Noise Models with Hidden Variables

We study causal discovery with hidden variables when causes modulate both the mean and the spread of their effects, a location-scale noise model; existing functional-model methods for hidden variables assume additive noise. We introduce LSNM-UV, a two-level location-scale model in which the hidden parents of a variable act inside the scale set by its observed parents, so that observed-parent effects can be divided out: every hidden effect collapses into an effective noise whose dependence structure encodes unobserved backdoor and causal paths. The resulting residual-independence patterns identify directed edges, non-edges and hidden-variable pairs, and identify bow-free acyclic directed mixed graphs. Under an explicit search-coverage condition, an oracle answering residual-independence queries at the population level makes the LSNM-UV search sound and complete. A thresholded HSIC statistic on fitted residuals answers every query on the oracle path correctly with probability tending to one under four explicit conditions on admissible population targets, targeting of independence witnesses, consistency of the empirical dependence statistic and a vanishing threshold, which gives graph consistency for a fixed number of variables; a fixed test level need not have this property. Controlled simulations diagnose the statistical conditions and their failure modes, an oracle-driven search shows that correct answers yield the correct graph, and a ten-node benchmark and the NetSim fMRI benchmark with hidden regions show the method at finite samples.