On the Identifiability of Poisson Branching Structural Causal Model Under Latent Confounding

Abstract

Causal discovery from observational count data poses unique challenges, particularly when the data exhibit inherent branching structures, such as an upstream ad impression event triggering a downstream purchase event with certain probability. Such branching dynamics are naturally modeled by thinning operators (for branching) and an independent Poisson distribution (for exogenous noise), constituting a Poisson Branching Structural Causal Model (PB-SCM). However, existing approaches based on PB-SCM rely on the restrictive assumption of causal sufficiency, failing to account for ubiquitous latent confounders. In this work, we propose a Latent Confounding Poisson Branching Structural Causal Model (LC-PB-SCM) to bridge this gap. We leverage Probability Generating Function (PGF) to characterize the complex dependencies introduced by latent confounding. Then, we establish a Trie representation theorem that maps the branching structure to algebraic properties of PGF monomials. Based on local PGF, we establish a complete identifiability condition for local 3-variables covering all causal patterns distinguishable up to monomial equivalence. Finally, we propose a practical algorithm to learn causal structures under latent confounding and demonstrate its effectiveness through experiments on both synthetic and real-world datasets.