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

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.

Many real-world events are measured in counts: the number of ads shown, clicks received, or subsequent purchases made. To understand cause and effect from such data, we need methods that recognize how one event can directly trigger another. This paper addresses this challenge specifically for count data, where hidden, unrecorded factors might influence multiple observed events at the same time. For example, a marketing campaign that is not recorded in the data might affect both ad impressions and purchases, muddying the true cause-and-effect relationship between them. We introduce a new method designed specifically for these scenarios, demonstrating how to extract accurate causal information from the data's patterns. The main idea is to translate these patterns into a mathematical summary that reveals which allows us to identify which causal explanations are distinguishable from one another. This leads to a practical method for learning causal structure even when some underlying causes are unobserved. Experiments on simulated and real data verify that the method can recover meaningful causal relationships in settings where existing approaches fail.