Cardinality Bounds and Parameterization for Latent Variable Graphical Models

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Claim: any latent discrete variable can be assumed finite without restricting the marginal model, and tighter Carathéodory's theorem bounds follow by measuring which marginal dimensions each latent actually affects. Method: use nested Markov parameters to drop irrelevant dimensions, choose which counterfactuals to model, and apply a hybrid parameterization (tasking some latents, using DAG parameters for the rest). Results: exponential bounds like 2^12 (4096) for K=11 drop to 48 for central latents; polynomial-system checks for causal compatibility then need far fewer unidentified parameters (e.g. 25→23 vs hybrid 9 total/7 unidentified).

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1. 0:00Start
2. 0:16Parameterization and the Cardinality Bonds for Latent Variable Graphic Models
3. 1:14The Causal Compatibility Question
4. 2:29Arbitrary Models for Discrete Data
5. 4:22Chirotheodore's Theorem
6. 10:04Equality Constraints Reduce Model Dimension
7. 10:29Four Equality Constraints
8. 12:39Nested Markov Parameterization
9. 13:38Nested Markov Parameters
10. 31:01Number of Unidentified Parameters
11. 31:47Hybrid Parameterization
12. 32:04The Hybrid Parametrizations
13. 37:23Conclusions