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Scores for Learning Discrete Causal Graphs with Unobserved Confounders
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Proposes a Bayesian score (SWBIC, built on Watanabe's WBIC and bounds on exogenous cardinality) for discrete causal graphs with unobserved confounders that captures equality and inequality constraints, and implements it with MCMC and greedy search.
Problem
Structure learning of causal graphs with unobserved confounders from discrete data is limited by conditional-independence-based equivalence classes, which miss equality and inequality constraints (e.g., Verma constraint, IV inequality); BIC is invalid for these singular models.
Approach
Parameterize arbitrary causal graphs with discrete exogenous variables of bounded cardinality so that all observational constraints are captured, then score graphs with a tempered (WBIC-style) posterior expectation of the log-likelihood estimated by MCMC, which is sound, parsimonious, decomposable over c-components and score equivalent.
Key results
- SWBIC correctly scores IV graph over unconstrained DAG with inequality constraint, where SBIC1 incorrectly favors the DAG in most cases.
- SWBIC is shown to be score equivalent and decomposable over c-components, with theoretical consistency under Assumptions 1-2.
- Greedy search with SWBIC (GS-SWBIC) is reported as competitive on Sachs and Lung cancer benchmarks with omitted variables.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SWBIC instead of SBIC1 | Causal graph scoring on synthetic IV comparisons · Synthetic IV vs unconstrained DAG | proportion of experiments ranking the IV graph better | Helped |
| Use SWBIC instead of SBIC2 | Causal graph scoring on synthetic IV comparisons · Synthetic IV vs unconstrained graph with equal edge and parameter counts | proportion of experiments ranking the IV graph better | Helped |
| Use SWBIC instead of SBIC1 or SBIC2 | Causal graph scoring on synthetic IV comparisons · Synthetic IV vs unconstrained graph with fewer edges and parameters | proportion of experiments ranking the IV graph better | Helped |
| Use SWBIC to score equivalent graph structures | Causal graph scoring · Synthetic IV vs equivalent graph | score equivalence | Helped |
| Search arbitrary causal graphs with GS-SWBIC rather than restricting search to existing graph classes | Causal structure learning · Synthetic IV and Verma models | Bayes factor versus optimal DAG | Helped |
| Search arbitrary causal graphs with GS-SWBIC | Causal structure learning · Synthetic front-door model | Bayes factor versus optimal DAG | No clear effect |
Surprising findings
- Standard BIC penalties mis-score graphs with inequality constraints, favoring unconstrained DAGs with fewer parameters.
- The front-door graph is empirically indistinguishable from a fully connected DAG in the Bayes factor analysis.
Limitations
- Restricted to discrete observed variables; continuous endogenous variables would need parametric assumptions.
- Learning coefficient is generally unknown and hard to evaluate, so the score relies on MCMC approximation.
- Evaluation is on simulations and a few small benchmarks.
Experimental setup
Datasets: synthetic data from SCMs over IV, Verma, front-door graphs Sachs Lung cancer
Benchmarks: synthetic IV/Verma/front-door graph variations Sachs Lung cancer
Metrics: proportion of experiments where true graph scored better Bayes factor vs optimal DAG
Baselines: SBIC1 SBIC2 DCD GES GSMAG
Builds on: BIC Watanabe's WBIC / singular learning theory Verma constraint Zhang, Jin, Bareinboim 2022 cardinality bounds GES GSMAG
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