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Identification of Causal Structure with Latent Variables Based on Higher Order Cumulants
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Proposes higher-order-cumulant identifiability conditions to detect whether a causal edge exists between two observed variables sharing one latent confounder in a non-Gaussian linear model, and to orient that edge, evaluated on synthetic data.
Problem
Causal discovery with latent variables: fully identifying the structure where two observed variables are both influenced by one latent variable and may have a directed edge between them, which prior methods (FCI, GIN, RCD, LvLiNGAM) do not identify.
Approach
Specific combinations of higher-order joint cumulants of two observed variables satisfy a tetrad-like constraint (C_{i+1,j+1}^2 = C_{i,j+2} C_{i+2,j}) iff no directed edge exists; with an edge, fifth-order cumulants give a quadratic for the mixing-coefficient ratios, and a third-order cumulant asymmetry criterion determines direction.
Key results
- In Case 2 (edge exists, linear, exponential noise, n=100000) the proposed method reaches about 0.92 accuracy vs 0.65 for LvLiNGAM and 0.07 for LiNGAM.
- In Case 1 (no edge) LvLiNGAM and LiNGAM score 0.00 accuracy while the proposed method scores 0.55-0.75 across distributions and sample sizes.
- On gamma-noise data the method is outperformed by LvLiNGAM in the nonlinear setting (e.g., 0.34-0.38 vs 0.43-0.54).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use higher-order cumulant criteria to detect edges under a shared latent variable | Proposed method · Case 1, synthetic linear data | accuracy: 0.64; 0.75; 0.60; 0.69 (Exp) | Helped |
| Use higher-order cumulants and asymmetry criterion for causal direction | Proposed method · Case 2, synthetic linear data | accuracy: 0.79; 0.88; 0.91; 0.92 (Exp) | Helped |
| Apply the method to nonlinear causal relationships | Proposed method · Case 2, nonlinear Exp data, n=100000 | accuracy: 0.84 | Helped |
| Apply the method to nonlinear causal relationships | Proposed method · Case 2, nonlinear Gamma data | accuracy: 0.38; 0.36; 0.34; 0.37 | Hurt |
Surprising findings
- Under the nonlinear generation process with gamma noise, the proposed method underperforms LvLiNGAM, which the authors attribute to larger nonlinear distortion of the gamma variables.
Limitations
- Type I error around 0.3 at alpha=0.0001 and accuracy only 0.55-0.75 in the no-edge case, with high sensitivity to sample size.
- Evaluated only on two-variable synthetic graphs; extension to high-dimensional settings is left open.
- Requires non-Gaussian noise and a linear (or only slightly nonlinear) model; performance degrades with gamma noise under nonlinearity.
Experimental setup
Datasets: synthetic linear non-Gaussian data (exponential, gamma, Gumbel noise)
Benchmarks: synthetic two-variable latent confounder graphs (Case 1 and Case 2)
Metrics: accuracy type I error type II error
Baselines: LiNGAM ANM LvLiNGAM
Builds on: LiNGAM LvLiNGAM overcomplete ICA tetrad constraint
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