Index: Bayesian Causal Inference Methods

Bayesian Causal Inference Methods

Routing Summary

This folder covers the Bayesian approach to causal inference: the likelihood factorization, prior independence assumption, outcome model specification, and the role of the propensity score. Contains 3 notes.

• Need the core Bayesian CI factorization, prior independence (Assumption 3.2), SATE vs. MATE? → General Structure of Bayesian CI
• Need BART, Gaussian Process, BCF, regularization-induced confounding? → Bayesian Outcome Models
• Need the three strategies for the propensity score in Bayesian analysis? → Propensity Score in Bayesian CI

Concept Map

Concept	Note	Type	Depends On	Key Result
Full-data likelihood factorization	General Structure of Bayesian CI	definition	Potential Outcomes Framework	Three terms: assignment, outcome, covariate
Prior independence (Assumption 3.2)	General Structure of Bayesian CI	definition	—	$p({\theta }_Z,{\theta }_Y,{\theta }_X)$ factorizes; propensity score ignorable
Prior dogmatism	General Structure of Bayesian CI	concept	Prior independence	In high-$p$, Assumption 3.2 acts as strongly informative prior
Bayesian identifiability	General Structure of Bayesian CI	concept	—	All parameters have posteriors; transparent parametrization needed
BART outcome model	Bayesian Outcome Models	definition	General Structure	S/T-learner; outperforms random forests empirically
Bayesian Causal Forest (BCF)	Bayesian Outcome Models	definition	BART	Separates prognostic + treatment effect functions
Regularization-induced confounding	Bayesian Outcome Models	concept	Prior independence	Standard regularization priors bias causal estimates in high-$p$
Propensity score as covariate	Propensity Score in Bayesian CI	concept	General Structure	Double robustness; BCF strategy
Dependent priors	Propensity Score in Bayesian CI	concept	General Structure	Joint prior links ${\theta }_Z$ and ${\theta }_Y$
Posterior predictive p-value	Propensity Score in Bayesian CI	concept	General Structure	Plug posterior draws into doubly-robust estimator

Notes

• General Structure of Bayesian CI — CONTAINS: full-data likelihood factorization (definition), prior independence Assumption 3.2 (definition), propensity score ignorability, Bayesian identifiability, data augmentation for SATE, Example 3.1 (bivariate normal covariate adjustment), Bayesian bootstrap note
• Bayesian Outcome Models — CONTAINS: linear/BART/GP/BCF outcome model definitions, Example 4.1 (priors and overlap for CATE estimation), regularization-induced confounding warning, high-dimensional challenges
• Propensity Score in Bayesian CI — CONTAINS: three strategies (covariate, dependent priors, PPP), feedback problem explanation, Hájek IPW connection, summary comparison table

Sources

• Li et al. - 2022 - Bayesian causal inference a critical review.pdf — §3–5, pp. 5–13

See Also

• Foundations — prerequisite potential outcomes framework
• Sensitivity and Complex Mechanisms — extensions to non-ideal settings
• Bayesian Propensity Scores and IPW — existing vault note on Bayesian IPW (Heiss blog)
• Nonparametric Causal Inference — existing vault note on BART/non-parametric causal methods