Causal Discovery from Poisson Branching Structural Causal Model Using
  High-Order Cumulant with Path Analysis
Jie Qiao Yu Xiang Zhengming Chen Ruichu Cai Zhifeng Hao
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.