Doubly robust inference via calibration (2411.02771v2)

Abstract: Doubly robust estimators are widely used for estimating average treatment effects and other linear summaries of regression functions. While consistency requires only one of two nuisance functions to be estimated consistently, asymptotic normality typically require sufficiently fast convergence of both. In this work, we correct this mismatch: we show that calibrating the nuisance estimators within a doubly robust procedure yields doubly robust asymptotic normality for linear functionals. We introduce a general framework, calibrated debiased machine learning (calibrated DML), and propose a specific estimator that augments standard DML with a simple isotonic regression adjustment. Our theoretical analysis shows that the calibrated DML estimator remains asymptotically normal if either the regression or the Riesz representer of the functional is estimated sufficiently well, allowing the other to converge arbitrarily slowly or even inconsistently. We further propose a simple bootstrap method for constructing confidence intervals, enabling doubly robust inference without additional nuisance estimation. In a range of semi-synthetic benchmark datasets, calibrated DML reduces bias and improves coverage relative to standard DML. Our method can be integrated into existing DML pipelines by adding just a few lines of code to calibrate cross-fitted estimates via isotonic regression.

• The paper introduces C-DML estimators that use isotonic calibration to achieve doubly robust asymptotic linearity for linear functional inference.
• It details a construction combining cross-fitting, debiasing, and calibration to correct for slow or inconsistent nuisance parameter estimates.
• Empirical and theoretical results validate the method’s efficacy in enhancing robust causal inferences under weak estimation conditions.

Overview of "Automatic Doubly Robust Inference for Linear Functionals via Calibrated Debiased Machine Learning"

This paper introduces a nuanced approach for causal inference, specifically targeted at making inferences about estimands expressible as linear functionals of the outcome regression. The authors propose a novel class of debiased machine learning estimators, identified as calibrated debiased machine learning (C-DML) estimators, which are characterized by their doubly robust asymptotic linearity.

Key Contributions:

1. Calibration and Double Robustness: A central to the novel approach is the connection between calibration—typically a tool for prediction and classification—and achieving conditions for doubly robust asymptotic linearity. The C-DML estimator exploits this link by employing isotonic calibration to adjust nuisance function estimators, enhancing their robustness to slow or inconsistent estimations.
2. Construction of C-DML Estimators: The paper details the mathematical construction of a specific C-DML estimator that integrates cross-fitting, isotonic calibration, and debiased machine learning. By maintaining asymptotic linearity when either the outcome regression or the Riesz representer of the linear functional is accurately estimated, the C-DML enables valid statistical inferences despite potential deficiencies in the estimation of nuisance parameters.
3. Empirical and Theoretical Results: The empirical results, combined with theoretical insights, substantiate the use of C-DML estimators in mitigating bias when nuisance functions are estimated inconsistently or at suboptimal rates.

Theoretical Implications:

The work significantly advances the understanding of robust causal inference by highlighting the role of calibration. By ensuring that debiased machine learning estimators are doubly robust asymptotically linear, this approach does not only provide a safeguard against slow convergence of nuisance function estimators but also ensures the robustness of statistical inference.

Practical Applications:

Given the robustness properties, C-DML estimators can significantly impact various domains where causal inference is crucial, such as epidemiology, economics, and social sciences. The ability to produce valid inference under weak conditions on nuisance estimators can enhance the reliability of findings in real-world applications where data imperfections are prevalent.

Future Directions:

Future research may explore extending the C-DML framework to other functionals beyond linear ones and investigating its applicability in diverse complex data structures. There's potential for further integration of machine learning advancements with causal inference methods to improve the scalability and adaptability of this approach.

In conclusion, this paper makes substantial contributions to causal inference through its introduction of C-DML estimators, offering a robust, practical, and theoretically sound approach to address the challenges posed by nuisance function estimation. This work lays the foundation for future exploration into more generalized and flexible inference frameworks within machine learning-informed causal inference methodologies.