Causal DAG Identification for Count Data via Poisson Thinning Structural Equation Models

Penggang Gao; Ming Cai; Hisayuki Hara

stat.MESep 5, 2026

Causal DAG Identification for Count Data via Poisson Thinning Structural Equation Models

Authors: Penggang Gao, Ming Cai, Hisayuki Hara

Organizations: Graduate School of Informatics Kyoto University Kyoto, 606-8501, Japan · Institute for Liberal Arts and Sciences Kyoto University Kyoto, 606-8501, Japan

Abstract

Count-valued variables arise in many scientific and applied settings, yet explicit structural models that allow full identification of causal DAGs from observational data remain limited. The Poisson branching structural causal model (PB-SCM) provides a count-valued analogue of linear structural equation models using binomial thinning and independent Poisson exogenous variables, but its causal DAG is generally only partially identifiable. Building on this framework, we propose the Poisson thinning structural equation model (PT-SEM), which replaces binomial thinning in PB-SCM with Poisson thinning and allows node-wise exogenous distributions from diverse count-distribution families. Under node-wise regularity conditions, we establish identifiability of the causal DAG, the thinning coefficients, and the node-wise exogenous distributions. The same identification analysis extends to binomial thinning, yielding full identifiability whenever every nonsink has non-Poisson exogenous noise. We further develop a structure learning algorithm that optimizes, via dynamic programming, a BIC score based on local likelihoods evaluated at plug-in moment estimates, and establish its consistency for DAG selection. Simulations demonstrate favorable performance in DAG recovery and thinning-coefficient estimation, and a real-data application illustrates the practical utility of PT-SEM.

Figures & tables

Family	Parameter setting
Poisson	$\lambda_{i}\sim\operatorname{Uniform}(2,10)$
Negative binomial	$r_{i}\sim\operatorname{Uniform}(2,10)$ , $p_{i}\sim\operatorname{Uniform}(0.15,0.85)$
Zero-inflated Poisson	$\lambda_{i}\sim\operatorname{Uniform}(2,10)$ , $\rho_{i}\sim\operatorname{Uniform}(0.15,0.85)$
Geometric	$\mu_{i}\sim\operatorname{Uniform}(2,10)$ , $p_{i}=(1+\mu_{i})^{-1}$
Binomial	$n_{i}$ is drawn uniformly from $\{2,\ldots,20\}$ , $p_{i}\sim\operatorname{Uniform}(0.15,0.85)$
Bernoulli	$p_{i}\sim\operatorname{Uniform}(0.15,0.85)$

Table 1: Parameter settings for the exogenous distributions.

Item	Setting
Dimension sweep	$d\in\{4,5,6,7,8,9,10\}$ , $N=3200$ , $\bar{k}_{\mathrm{in}}=1.5$
Sample size sweep	$N\in\{100,200,400,800,1600,3200,6400,10000\}$ , $d=8$ , $\bar{k}_{\mathrm{in}}=1.5$
Average in-degree sweep	$\bar{k}_{\mathrm{in}}\in\{1.0,1.5,2.0,2.5,3.0\}$ , $d=8$ , $N=3200$

Table 2: Regime and sweep settings

Family	Moment inversion $H_{r}(m,v)$	Mass-ratio order
Poisson	$\lambda=m$	$O((1+x)^{h})$
Negative binomial	$r=m^{2}/(v-m),\ p=m/v$	$O(1)$
Zero-inflated Poisson	$\rho=(v-m)/(v-m+m^{2}),\ \lambda=(v-m+m^{2})/m$	$O((1+x)^{h})$
Geometric	$p=(1+m)^{-1}$	$O(1)$
Binomial	$n=\operatorname{round}\{m^{2}/(m-v)\},\ p=m/n$	$O(1)$
Bernoulli	$p=m$	$O(1)$

Table 3: Moment inversions and mass-ratio bounds for the candidate exogenous families.

Figure 1: Directed-edge F1 in the all-Poisson setting.

Figure 2: Directed-edge F1 in the

\mathcal{M}

setting.

Figure 3: Thinning-coefficient MAPE over correctly recovered edges in the

\mathcal{M}

setting.

Figure 5: Prespecified reference DAG for the NBA event counts.

Method	Skeleton recovery			Directed-edge recovery
	Prec.	Rec.	F1	Prec.	Rec.	F1
Proposed DP-BIC	1.000	1.000	1.000	1.000	1.000	1.000
ODS	0.800	1.000	0.889	0.800	1.000	0.889
PC-RCIT	0.619	1.000	0.764	0.148	0.200	0.169
PB-SCM	0.900	0.575	0.688	0.650	0.300	0.399
PB-SCM-PGF	0.719	0.925	0.806	0.423	0.450	0.423

Table 4: Average structural-recovery performance over the ten NBA seasons.

Figure 6: Season-wise PT-SEM estimates for the NBA data.

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