However, the Bayesian network and ordinal modeling ignore the inherent branching structure among the counting relationship, which is frequently encountered in real world (Weiß [2018]).

Causal discovery from observational data, especially for count data, is essential across scientific and industrial contexts, such as biology, economics, and network operation maintenance. However, they overlook the inherent branching structures that are frequently encountered, e.g., a browsing event might trigger an adding cart or purchasing event. This can be modeled by a binomial thinning operator (for branching) and an additive independent Poisson distribution (for noising), known as Poisson Branching Structure Causal Model (PB-SCM). There is a provably sound cumulant-based causal discovery method that allows the identification of the causal structure under a branching structure. However, we show that there still remains a gap in that there exist causal directions that are identifiable while the algorithm fails to identify them. In this work, we address this gap by exploring the identifiability of PB-SCM using the Probability Generating Function (PGF).

Count data naturally arise in many fields, such as finance, neuroscience, and epidemiology, and discovering causal structure among count data is a crucial task in various scientific and industrial scenarios. One of the most common characteristics of count data is the inherent branching structure described by a binomial thinning operator and an independent Poisson distribution that captures both branching and noise. For instance, in a population count scenario, mortality and immigration contribute to the count, where survival follows a Bernoulli distribution, and immigration follows a Poisson distribution. However, causal discovery from such data is challenging due to the non-identifiability issue: a single causal pair is Markov equivalent, i.e., $X\rightarrow Y$ and $Y\rightarrow X$ are distributed equivalent. Fortunately, in this work, we found that the causal order from $X$ to its child $Y$ is identifiable if $X$ is a root vertex and has at least two directed paths to $Y$, or the ancestor of $X$ with the most directed path to $X$ has a directed path to $Y$ without passing $X$. Specifically, we propose a Poisson Branching Structure Causal Model (PB-SCM) and perform a path analysis on PB-SCM using high-order cumulants. Theoretical results establish the connection between the path and cumulant and demonstrate that the path information can be obtained from the cumulant. With the path information, causal order is identifiable under some graphical conditions. A practical algorithm for learning causal structure under PB-SCM is proposed and the experiments demonstrate and verify the effectiveness of the proposed method.