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Discussion on "Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects" by Hahn, Murray and Carvalho
Liangyuan Hu
TL;DR
The paper studies how the BCF model supports causal inference and clarifies two roles of propensity scores: their distinction from Bayesian propensity-score updating and their connection to confounding under targeted selection. It explains how propensity-score inclusion can reduce bias when ignorability is violated and identifies extensions needed for ATT and multiple-treatment settings.
Problem
The discussion addresses the need to distinguish BCF’s use of estimated propensity scores from Bayesian propensity-score updating and to clarify propensity scores’ role in outcome-based causal-effect estimation.
Method
The paper discusses BCF’s propensity-score formulation, connects propensity-score adjustment to the confounding function, and outlines possible methodological extensions.
Results
The discussion concludes that propensity scores are important for bias reduction under strong targeted selection but have a diminished role when targeted selection is absent.
Takeaways & Limitations
BCF avoids the described Bayesian model-feedback issue and offers a framework whose propensity-score contribution is most consequential when ignorability violations are substantial.
Takeaways & Limitations
BCF does not appear readily implementable for ATT estimation and requires extension to compare more than two active treatments.
Abstract
from arXiv · showhide
Hahn et al. (2020) offers an extensive study to explicate and evaluate the performance of the BCF model in different settings and provides a detailed discussion about its utility in causal inference. It is a welcomed addition to the causal machine learning literature. I will emphasize the contribution of the BCF model to the field of causal inference through discussions on two topics: 1) the difference between the PS in the BCF model and the Bayesian PS in a Bayesian updating approach, 2) an alternative exposition of the role of the PS in outcome modeling based methods for the estimation of causal effects. I will conclude with comments on avenues for future research involving BCF that will be important and much needed in the era of Big data.
1 Distinction from the Bayesian propensity scores
The BCF model’s estimated propensity score is an externally estimated covariate used for tree splitting, unlike Bayesian propensity scores jointly updated with outcome information. This separation avoids the Bayesian model-feedback issue described for causal inference.
- 1 Distinction from the Bayesian propensity scores: Bayesian propensity-score approaches combine propensity-score modeling and Bayesian outcome inference in a single updating procedure.Prior work has shown that outcome-to-propensity-score feedback can distort causal-effect inference.
- 1 Distinction from the Bayesian propensity scores: BCF includes the estimated propensity score as an additional tree-splitting dimension rather than updating it with outcome information.The model places an independent BART prior over the response function using the estimated propensity score as a covariate.
- 1 Distinction from the Bayesian propensity scores: Because BCF does not update or contaminate the estimated propensity score with outcome information, it avoids the inference issue caused by Bayesian model feedback.The distinction concerns how propensity-score information enters the model, not whether it is used in outcome modeling.
2 The role of the propensity score: connections to the confounding function
The propensity score helps explain and reduce bias from targeted selection through its connection to the confounding function. Its importance increases with the degree of ignorability violation and diminishes when targeted selection is absent.
- 2 The role of the propensity score: connections to the confounding function: Under targeted selection, treatment assignment depends on the untreated outcome, so treated individuals can have higher potential untreated outcomes than untreated individuals.This pattern violates the ignorability assumption and produces biased causal-effect estimates.
- 2 The role of the propensity score: connections to the confounding function: The confounding function c(z, x) represents the difference in potential outcomes between treated and untreated individuals conditional on covariates.It equals zero under ignorability and can be positive for untreated potential outcomes under targeted selection.
- 2 The role of the propensity score: connections to the confounding function: Adjusting outcomes with the propensity score and confounding function yields an unbiased effect estimate by differencing adjusted conditional expectations across treatment groups.Equation (2.1) defines the adjusted outcome used for this comparison.
- 2 The role of the propensity score: connections to the confounding function: When targeted selection is strong, the confounding function deviates substantially from zero and the estimated propensity score becomes important for bias reduction.When targeted selection is absent, the confounding function is near zero and the propensity score’s role is diminished.
3 Final thought on possible extensions
The discussion identifies extensions needed for emerging causal-inference applications, especially ATT estimation and settings with more than two treatments.
- 3 Final thought on possible extensions: The BCF model does not appear readily implementable for estimating the average treatment effect on the treated, although ps-BART can be applied to ATT estimation.This marks ATT estimation as an unresolved implementation need for BCF.
- 3 Final thought on possible extensions: Large-scale biomedical data often involve treatment regimens with more than two treatments, motivating extensions of BCF to multiple-treatment comparisons.The cited discussion reports that BART-based methods generally performed best in one multiple-treatment investigation.