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On the Identifiability of Poisson Branching Structural Causal Model Using Probability Generating Function
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Derives a closed-form probability generating function for Poisson branching SCMs and uses local PGF rank tests to learn causal skeletons and orient edges, closing an identifiability gap left by cumulant-based methods.
Problem
Causal discovery from count data generated by Poisson branching structural causal models (binomial thinning plus Poisson noise), where the prior cumulant-based method leaves some identifiable causal directions unresolved.
Approach
Derives an exact closed-form PGF for PB-SCM where each term corresponds to a specific local path structure; local PGFs over small vertex subsets then allow adjacency, triangular, and collider tests via matrix rank conditions on PGF derivatives.
Key results
- On synthetic 10-vertex graphs with 30000 samples, the PGF method reaches F1 0.86 and SHD 6.40, versus 0.80 and 13.00 for the cumulant baseline.
- Across average in-degree 2.0 to 3.5, the PGF method's F1 rises from 0.74 to 0.89 and SHD falls from 9.67 to 5.27.
- On the football events data it orients six causal directions plus one adjacency consistent with theory; on the shopping data it recovers the Impressions-Clicks adjacency and the other two directions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 2.0) · Synthetic (avg. in-degree 2.0) | F1: 0.73 ± 0.03 -> 0.74 ± 0.05 | Mixed |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 2.5) · Synthetic (avg. in-degree 2.5) | F1: 0.77 ± 0.02 -> 0.81 ± 0.07 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 3.0) · Synthetic (avg. in-degree 3.0) | F1: 0.80 ± 0.04 -> 0.86 ± 0.03 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 3.5) · Synthetic (avg. in-degree 3.5) | F1: 0.83 ± 0.03 -> 0.89 ± 0.04 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 2.0) · Synthetic (avg. in-degree 2.0) | SHD: 13.40 ± 1.28 -> 9.67 ± 1.99 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 2.5) · Synthetic (avg. in-degree 2.5) | SHD: 14.10 ± 1.51 -> 8.33 ± 2.28 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 3.0) · Synthetic (avg. in-degree 3.0) | SHD: 13.00 ± 2.37 -> 6.40 ± 1.68 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (avg. in-degree 3.5) · Synthetic (avg. in-degree 3.5) | SHD: 13.20 ± 2.23 -> 5.27 ± 1.39 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (sample size 5000) · Synthetic (sample size 5000) | F1: 0.72 ± 0.04 -> 0.75 ± 0.09 | Mixed |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (sample size 15000) · Synthetic (sample size 15000) | F1: 0.78 ± 0.02 -> 0.82 ± 0.04 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (sample size 30000) · Synthetic (sample size 30000) | F1: 0.80 ± 0.04 -> 0.86 ± 0.03 | Helped |
| Use the PGF-based method instead of Cumulant | Synthetic causal discovery (sample size 50000) · Synthetic (sample size 50000) | F1: 0.80 ± 0.03 -> 0.87 ± 0.04 | Helped |
Surprising findings
- The cumulant-based method fails to identify X1 -> X2 <- X3 in a 4-vertex graph, even though the direction is identifiable in principle.
Limitations
- Explicit estimation of the probability generating function does not scale well with the number of nodes.
- Relies on causal faithfulness and causal sufficiency assumptions, which may not hold in practice.
- Some directions (e.g., in triangular structures without an in-degree-2 vertex) remain unidentifiable.
Experimental setup
Datasets: Football Events dataset (Kaggle) Shopping mall paid search campaign dataset (Kaggle)
Benchmarks: synthetic PB-SCM graphs (3, 4, 5, 10 vertices) Football Events Shopping mall paid search campaign
Metrics: F1 SHD
Baselines: Cumulant PC GES OCD
Builds on: Poisson branching structural causal model cumulant-based causal discovery for PB-SCM PC algorithm bootstrap
Closest papers by meaning
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