PoET: Test-Integrated Nonparametric Poisson DAG Learning via Equidispersion Testing

Poisson DAG models enable causal discovery for count data through a mean–variance asymmetry: conditioning on all parents yields equidispersion, whereas omitted parents induce overdispersion. Existing ranking-based methods exploit this asymmetry to construct a causal ordering by repeatedly selecting the node with the smallest overdispersion, without testing whether the selected node is statistically compatible with equidispersion. These methods also face difficulties in nonparametric conditional-mean estimation or avoid them by prespecifying nodewise link functions, which limits modeling flexibility. We propose PoET, a test-integrated framework for Poisson DAG learning that replaces score ranking with test-based layer eligibility. Its new cross-fitted tests for conditional uncorrelatedness and equidispersion use Conway–Maxwell–Poisson (CMP)-kernel smoothing to estimate conditional means nonparametrically, without prespecified nodewise link functions. To mitigate the conditioning burden of nonparametric smoothing, conditional-uncorrelatedness tests screen parent candidates to reduce conditioning dimension, while equidispersion tests guide layer assignment and parent recovery. If no candidate is eligible for layer assignment, PoET stops rather than forcing further graph decisions. We establish graph-recovery consistency under correct specification and consistent violation detection under local misspecification. Simulations show competitive recovery under heterogeneous nonlinear mechanisms and reliable detection of dispersion violations. NBA and Lending Club analyses illustrate, respectively, test-based incompatibility detection and complete layerwise learning without prescribed links.