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A Generic Family of Graphical Models: Diversity, Efficiency, and Heterogeneity
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Proposes the marginally recoverable parametric family and an EM-MMLE estimator that uses pairwise (2-D marginal) likelihoods to estimate covariance/precision matrices for Gaussian, PLN, and binary latent Gaussian copula graphical models, including mixtures for heterogeneous populations.
Problem
Network inference for discrete and heterogeneous data (e.g., scRNA-seq counts) is hard because maximum likelihood for models like Poisson log-normal and latent Gaussian copulas requires intractable high-dimensional integrals.
Approach
Define a distribution family whose parameters are fully identified by its one- and two-dimensional marginals, so the covariance can be estimated by maximizing low-dimensional marginal likelihoods (MMLE), extended to mixtures via EM (EM-MMLE) with D-trace precision estimation.
Key results
- On simulated MPLN random graphs (n=3000, p=300, low mixing, low zero-proportion), EM-MMLE achieves AUPR 0.96 versus VMPLN's 0.75, outperforming VMPLN by about 28%.
- On the blocked random graph (n=3000, p=300, low zero-proportion), EM-MMLE gets AUPR 0.84 (high mixing) and 0.94 (low mixing) versus PLNet's 0.73 and 0.87.
- On binary data, MMLE's estimation error for Sigma nearly matches Fan et al.'s rank-based method and is far below AMLE, with the Oracle as lower bound.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EM-MMLE instead of PLNet for network inference | MPLN random-graph simulations, n = 3000, p = 300, low mixing, low zero-proportion · MPLN random graphs | AUPR: 0.91 (0.052) -> 0.96 (0.008) | Helped |
| Use EM-MMLE instead of PLNet for network inference | MPLN random-graph simulations, n = 3000, p = 100, low mixing, low zero-proportion · MPLN random graphs | AUPR: 0.95 (0.047) -> 0.99 (0.006) | Helped |
| Use MMLE instead of Fan et al.'s estimator | Binary data, p = 50, n = 200 · Binary data | Frobenius error of covariance estimate: 6.375 (0.15) -> 6.343 (0.15) | No clear effect |
| Use MMLE instead of AMLE | Binary data, p = 50, n = 200 · Binary data | Frobenius error of covariance estimate: 31.254 (0.11) -> 6.343 (0.15) | Helped |
| Use MMLE instead of the Oracle estimator | Binary data, p = 50, n = 200 · Binary data | Frobenius error of covariance estimate: 3.599 (0.02) -> 6.343 (0.15) | Hurt |
| Use MMLE instead of Fan et al.'s estimator | Binary data, p = 200, n = 3000 · Binary data | Frobenius error of precision estimate: 2.33 (0.04) -> 2.332 (0.04) | No clear effect |
| Use MMLE instead of AMLE | Binary data, p = 200, n = 3000 · Binary data | Frobenius error of precision estimate: 2.871 (0.04) -> 2.332 (0.04) | Helped |
| Use MMLE instead of the Oracle estimator | Binary data, p = 200, n = 3000 · Binary data | Frobenius error of precision estimate: 1.045 (0.03) -> 2.332 (0.04) | Hurt |
| Use EM-MMLE for mixed-count network inference | MPLN simulations · MPLN simulated scenarios | AUPR | Helped |
| Use EM-MMLE for gene regulatory network inference | scRNA-seq dataset · scRNA-seq cell types | AUPRC ratio | Helped |
Limitations
- Pairwise/marginal estimators are not guaranteed to be efficient relative to full likelihood; the paper notes the restriction to 1-D and 2-D marginal structure
- Theoretical guarantees rely on conditions (bounded parameters, identifiability) and the consistency result requires p not growing faster than exp(n)
- Real-data evaluation relies on silver standards built from public databases rather than gold-standard networks
Experimental setup
Datasets: Zheng et al. 2017 scRNA-seq dataset (6,952 cells) simulated MPLN data simulated binary latent Gaussian copula data
Benchmarks: Zheng et al. 2017 scRNA-seq silver standards
Metrics: AUPR Frobenius norm estimation error AUPRC ratio
Baselines: PLNet VMPLN Glasso Fan et al. rank-based estimator AMLE PPCOR GENIE3
Builds on: Poisson log-normal model D-trace loss composite (pairwise) likelihood Gaussian copula PLNet
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