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

• Explicit count-valued SEMs with full DAG identifiability remain limited, while PB-SCM generally identifies only part of the DAG and restricts coefficients and exogenous noise.Other count-DAG approaches derive identifiability from restrictions on node-wise conditional distributions.Source: §1 · p. 2
• PT-SEM replaces binomial with Poisson thinning, allowing coefficients in [0, ∞) and mutually independent exogenous distributions from different families.Poisson offspring permit one-to-many reproduction, and coefficients represent mean offspring counts rather than probabilities.Source: §1 · p. 2
• Under node-wise regularity conditions, the observational distribution uniquely determines the PT-SEM DAG, thinning coefficients, and exogenous distributions.The same analysis yields full identifiability for binomial thinning when every nonsink has non-Poisson exogenous noise.Source: §1 · p. 2
• A decomposable plug-in BIC uses moment-based local estimates and candidate exogenous families, with exact subset dynamic programming and consistency for DAG and family selection.This avoids repeated numerical maximization of local likelihoods during structure search.Source: §1 · pp. 2–3

2.1 Basic facts on SEMs and Poisson Thinning

• PB-SCM combines binomial thinning with independent Poisson exogenous noise, restricting coefficients to (0, 1] and generally leaving the DAG only partially identifiable.These restrictions limit the model's expressive power.Source: §2.1 · pp. 3–4
• Poisson thinning is a generalized thinning operator with Poisson offspring, unlike classical binomial thinning with Bernoulli offspring.Each input unit independently produces a Poisson-distributed offspring count.Source: §2.1 · pp. 3–4

2.2 Poisson thinning structural equation model

• PT-SEM uses positive Poisson-thinning coefficients and mutually independent count-valued exogenous variables, without requiring a common parametric family.The coefficient αij is a direct effect on Xi's conditional mean and is not restricted to the unit interval.Source: §2.2 · pp. 4–5
• Under the minimal graph convention, an edge is present exactly when its coefficient is positive; acyclicity lets the equations define X recursively and the distribution factorize according to the DAG.The induced observational distribution is Markov with respect to G.Source: §2.2 · p. 5

3 Identifiability of PT-SEM

• The PT-SEM identifiability assumptions include independent exogenous variables with moment-generating functions finite near zero, and hold for several common count families.Examples include Poisson, zero-inflated Poisson, negative binomial, and geometric distributions.Source: §3 · p. 5

3.1 Identifiability in the Bivariate Case

• In the bivariate case, the forward PT-SEM direction is identifiable under (A1)–(A3) because no reverse PT-SEM representation generates the same joint distribution.The proof rules out the reverse representation using incompatible constraints on conditional moment-generating functions.Source: §3.1 · p. 6

3.2 Local sink characterization

• For a parent-closed subset, a node is a sink exactly when its conditional cumulant-generating function has the stated additive form; recursively removing sinks yields a topological order.The sink characterization is applied to the remaining induced subgraph at each step.Source: §3.2 · p. 7

3.3 Exact identifiability of PT-SEM

• Under (A1)–(A3), the observational distribution uniquely determines the PT-SEM DAG, coefficient matrix, and node-wise exogenous distributions.Covariance identities recover coefficients; their positive entries identify parents, and conditioning on preceding variables at zero recovers each exogenous distribution.Source: §3.3 · p. 8

3.4 Identifiability of PB-SCM

• For PB-SCM, a nonsink satisfying the sink characterization must have Poisson exogenous noise, so non-Poisson noise at every nonsink makes the DAG identifiable by recursive sink removal.Under that condition, the observational distribution uniquely determines the DAG, coefficient matrix, and node-wise exogenous distributions.Source: §3.4 · p. 9

4 Learning the causal structure in PT-SEM

• The learning task selects a DAG and a candidate exogenous family for each node, with the finite candidate set assumed to represent every node’s true exogenous distribution.The method also estimates the associated thinning coefficients.Source: §4 · p. 9

4.1 Likelihood under a fixed DAG and exogenous family assignment

• For a fixed DAG and family assignment, node-wise convolutional likelihoods and parameter counts decompose, but repeated numerical optimization across parent sets and families is costly.This motivates evaluating local likelihoods at moment-based estimates rather than repeatedly maximizing them numerically.Source: §4 · p. 9, §4.1 · p. 10

4.2 MoM-based parameter estimation

• Method-of-moments estimates use empirical means and covariances to estimate thinning coefficients and exogenous moments, which are mapped into candidate exogenous families.The estimates are constructed for each node and candidate parent set; the coefficient and exogenous-moment estimates do not depend on the candidate family.Source: §4.2 · pp. 10–11

4.3 Model evaluation via BIC

• The plug-in BIC consistently selects the DAG and exogenous-family assignment when the finite candidate collection represents each node’s true exogenous distribution and the stated regularity conditions hold.Redundant correctly specified models incur an additional log N penalty, while misspecified candidates lose O(N) in likelihood against an O(log N) dimension-penalty difference.Source: §4 · p. 9, §4.3 · pp. 12–13

4.4 Structure learning by exact dynamic programming

• Exact subset dynamic programming minimizes the decomposable plug-in BIC over candidate DAGs after local scores are computed.The procedure also stores coefficient estimates and minimizing family labels for node–parent-set pairs, then recovers the selected parameters and family assignment.Source: §4.4 · p. 13

5.1 Simulation design

• The simulations use PT-SEM-generated data with all-Poisson or node-varying exogenous families, two coefficient regimes, and sweeps over dimension, sample size, and average in-degree.Each configuration uses 100 independent replications.Source: §5 · p. 14, §5.1 · pp. 14–16

5.2 Results

• In the all-Poisson setting, Proposed DP-BIC consistently outperforms ODS and achieves DAG-recovery accuracy similar to Poisson-only DP.The difference from Poisson-only DP decreases as sample size increases.Source: Figure 1 · p. 16, §5.2 · p. 16
• In the M setting, Proposed DP-BIC performs nearly as well as Oracle DP across both coefficient regimes and outperforms Greedy-BIC and ODS in directed-edge F1.The gap from Greedy-BIC widens as average in-degree increases, while PC-RCIT yields lower scores and the PB-SCM methods perform substantially worse under the restricted regime.Source: §5.2 · p. 16, Figure 2 · p. 17
• Proposed DP-BIC closely tracks Oracle DP in thinning-coefficient MAPE, while Greedy-BIC deteriorates as average in-degree increases.MAPE decreases with sample size for Proposed DP-BIC and Oracle DP.Source: §5.2 · pp. 16–17, Figure 3 · p. 17
• Exogenous-family selection is generally accurate at sufficiently large sample sizes and more accurate in the restricted than extended regime.When sample size is around 100, DAG recovery remains nearly identical to Oracle DP despite less accurate family selection.Source: §5.2 · pp. 17–18

6 Real-World Event-Count Experiment

• The analysis used 95,808 team-quarter observations from the 2015–16 through 2024–25 NBA regular seasons, excluding overtime.The five counts were FOUL, FTA, FTM, PERS, and LOOSE.Source: §6 · p. 18
• The reference DAG places FOUL as a parent of PERS, LOOSE, and FTA, and FTA as a parent of FTM.Methods were fitted separately by season and evaluated against this prespecified graph.Source: §6 · pp. 18–19
• 1.000 precision, recall, and F1 for both skeleton and directed-edge recovery; Proposed DP-BIC recovered the complete reference DAG in every season.ODS had precision 0.800 and F1 0.889 for both recovery targets, while PC-RCIT, PB-SCM, and PB-SCM-PGF had lower directed-edge recovery.Source: Table 4 · p. 19, §6 · p. 19
• Selected exogenous families varied across nodes and seasons, while the recovered DAG matched the reference in every season.Figure 6 reports the estimated reference-edge coefficients and the selected family at each node by season.Source: §6 · p. 19, Figure 6 · p. 20
• FOUL→FTA estimates ranged from 1.187 to 1.297, averaged 1.238, and exceeded one in every season.The paper relates this magnitude to a qualifying foul generating multiple free-throw attempts.Source: §6 · p. 19, Figure 6 · p. 20

7 Conclusion

• PT-SEM uses Poisson thinning with coefficients in [0, ∞) and permits mutually independent, node-wise exogenous distributions without requiring a parametric family specification for identification.Under node-wise conditions, the observational distribution uniquely determines the DAG, thinning coefficient matrix, and exogenous distributions.Source: §7 · p. 20
• For binomial thinning, a nonsink satisfies the sink characterization only with Poisson exogenous noise, leaving the all-Poisson PB-SCM generally not fully identifiable.The PT-SEM identification proof uses a recursive sink characterization based on conditional cumulant generating functions.Source: §7 · p. 20
• The plug-in BIC combines local moment estimates with exact subset dynamic programming and is consistent for DAG and exogenous-family selection.It avoids repeated numerical maximization of local likelihoods.Source: §7 · p. 20
• Simulations favored DP-BIC in DAG recovery and coefficient estimation; the NBA analysis more closely matched reference relations than competing methods and produced stable estimates across seasons.The procedure selects exogenous families separately at each node.Source: §7 · p. 20
• The theory assumes causal sufficiency and acyclicity; extensions to latent confounding and feedback systems remain open.Scalable score optimization for larger graphs and models for mixed data types also remain open.Source: §7 · p. 20

A.1.1 Proof of Proposition 4

• In a topological ordering, every node’s parents occur earlier, and its value depends on preceding variables only through those parents.Its exogenous noise and incoming-edge Poisson offspring variables are independent of variables generated earlier.Source: §A.1.1 · p. 21
• Applying the chain rule in topological order yields the observational distribution’s factorization.The passage introduces this step for every x in the count-valued state space.Source: §A.1.1 · p. 21
• The resulting observational distribution factorizes according to G and is Markov.Source: §A.1.1 · p. 21

A.2.1 Proof of Lemma 5; A.2.4 Proof of Lemma 9

• Conditioning on zero-valued events yields linear and quadratic probability tilts of a nondegenerate baseline distribution, supporting the sink-characterization argument.The tilt coefficients have nonnegative constant and linear terms, with strictly positive linear and quadratic coefficients, respectively.Source: §A.2.1 · p. 22, §A.2.4 · p. 24

A.2.2 Proof of Lemma 6; A.2.3 Proof of Lemma 7; A.2.5 Proof of Proposition 10

• For PT-SEM, sinks satisfy the conditional cumulant-generating-function criterion, whereas applying it to a nonsink yields a contradiction.Conditional moment-generating functions under reverse representation have an exponential form; the resulting identities cannot hold for a nondegenerate baseline distribution.Source: §A.2.2 · p. 22, §A.2.3 · p. 23, §A.2.5 · p. 25

A.2.6 Proof of Lemma 11

• The covariance matrix on any nonempty set of vertices is positive definite, ensuring the covariance submatrices used in identification are invertible.The proof isolates an exogenous noise term independent of the other terms in a linear combination.Source: §A.2.6 · p. 25

A.2.7 Proof of Theorem 12

• The observational covariance identifies each node’s parent coefficients, while conditioning on zero predecessors identifies its exogenous distribution; together these determine the DAG and all node-wise parameters.The proof uses invertible predecessor covariance matrices and positive-probability zero-predecessor events.Source: §A.2.7 · p. 26

A.2.8 Proof of Proposition 13; A.2.9 Proof of Theorem 14

• The binomial-thinning sink criterion extends the identification proof, with a nonsink satisfying the criterion only in the Poisson-noise case.The proof also rules out a child coefficient of 1 for a nonsink under the criterion.Source: §A.2.8 · pp. 26–27, §A.2.9 · p. 28

A.3.1 Proof of Lemma 15

• The passage concludes that for every node k in V, induction establishes a positive t⋆ such that its moment-generating function MXk(t) is finite.Source: §A.3.1 · p. 28

A.3.2 Proof of Proposition 16

• The moment-based thinning-coefficient estimator has root-N consistency, and the discrete exogenous-family coordinates are recovered with probability tending to one.The proof transfers the rate to continuous parameter coordinates by the delta method and stacks the node-wise estimates.Source: §A.3.2 · pp. 28–29

A.3.3 Proof of Proposition 17

• Local score and Hessian terms admit polynomial envelopes near the true parameter, permitting differentiation under expectation and local likelihood expansions.The envelopes rely on the exponential-moment condition and mass-ratio bounds, including when the population thinning contribution is zero.Source: §A.3.3 · p. 29
• For a correctly specified candidate, the positive part of the likelihood difference is Op(1); in the true model, all edge coefficients are positive and continuous family coordinates are interior.These conditions yield a zero gradient for the true model’s population objective.Source: §A.3.3 · p. 30

A.3.4 Proof of Theorem 18

• A correctly specified candidate representation contains G0, and any extra edges have zero population coefficients, making its edge count strictly larger when its graph differs from G0.The exogenous distributions in correctly specified representations are the true node-wise distributions.Source: §A.3.4 · p. 30
• With probability tending to one, every candidate with G ≠ G0 or τ ∉ T0 has larger BICMoM than the true model, so the selected graph and family parameters are correct.The proof compares finitely many candidate models and concludes that Pr(Ĝ = G0, τ̂ ∈ T0) → 1.Source: §A.3.4 · p. 33

A.3.5 Computational complexity

• Subset-DP requires O(d^2 2^d) time, and overall space complexity is O(d^2 2^d).Retaining coefficient vectors for all node–parent-set pairs requires O(d^2 2^d) storage.Source: §A.3.5 · p. 33