Doubly Robust Inference for Linear Functionals Using Calibrated Debiased Machine Learning

Core Concepts

This research paper introduces a novel calibrated debiased machine learning (C-DML) framework for constructing doubly robust asymptotically linear estimators and confidence intervals for linear functionals of the outcome regression, even when one of the nuisance functions (outcome regression or Riesz representer) is estimated inconsistently or at arbitrarily slow rates.

Abstract

• Bibliographic Information: van der Laan, L., Luedtke, A., & Carone, M. (2024). Automatic doubly robust inference for linear functionals via calibrated debiased machine learning. arXiv preprint arXiv:2411.02771.

• Research Objective: This paper aims to develop a general framework for constructing doubly robust asymptotically linear (DRAL) estimators and confidence intervals for linear functionals of the outcome regression, addressing the limitations of existing methods that require consistent and sufficiently fast estimation of both the outcome regression and the Riesz representer.

• Methodology: The authors propose a novel calibrated debiased machine learning (C-DML) framework that leverages cross-fitting, isotonic calibration, and debiased machine learning estimation. The key idea is to orthogonalize the residuals of the nuisance function estimators to the projected residuals of the other nuisance function, effectively debiasing the cross-product remainder term that hinders asymptotic linearity in traditional methods.

• Key Findings: The paper demonstrates that C-DML estimators achieve asymptotic linearity even when one of the nuisance functions is estimated inconsistently or at arbitrarily slow rates, as long as the other nuisance function is consistently estimated at a sufficiently fast rate. The authors also propose a bootstrap-assisted approach for constructing doubly robust confidence intervals, eliminating the need for estimating additional nuisance functions.

• Main Conclusions: The C-DML framework provides a unified and computationally efficient approach for doubly robust inference on linear functionals of the outcome regression, applicable to a wide range of causal inference problems. The proposed estimators and confidence intervals are robust to misspecification or slow estimation of one of the nuisance functions, enhancing the reliability of causal effect estimation in practical settings.

• Significance: This research significantly advances the field of causal inference by providing a flexible and robust framework for estimating linear functionals of the outcome regression. The C-DML approach relaxes the stringent assumptions of existing methods, enabling more reliable causal effect estimation in situations where nuisance functions are difficult to estimate accurately.

• Limitations and Future Research: While the paper provides theoretical and empirical support for the C-DML framework, further investigation into its performance under different data-generating processes and with various machine learning algorithms is warranted. Future research could also explore extensions of the C-DML framework to handle more complex causal inference settings, such as time-varying treatments or high-dimensional data.

Stats

Quotes

Key Insights Distilled From

Automatic doubly robust inference for linear functionals via calibrated debiased machine learning

by Lars van der... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02771.pdf

Deeper Inquiries

How does the performance of C-DML compare to other doubly robust inference methods, such as targeted maximum likelihood estimation (TMLE), in high-dimensional settings with many covariates?

C-DML and TMLE are both doubly robust inference methods designed to provide valid inference for causal parameters even when one of the nuisance functions (outcome regression or Riesz representer) is estimated inconsistently or at a slow rate. However, they differ in their approach to achieving doubly robust asymptotic linearity (DRAL) and may exhibit different performance in high-dimensional settings.
C-DML, as described in the paper, leverages calibration techniques, such as isotonic calibration, to directly enforce empirical orthogonality conditions on the nuisance function estimators. This non-iterative approach can be advantageous in high-dimensional settings as it avoids the computational burden of iterative debiasing procedures often employed by TMLE. Additionally, C-DML benefits from using cross-fitting, which can further reduce bias and improve finite-sample performance, particularly when employing complex machine learning algorithms for nuisance function estimation.
TMLE, on the other hand, typically involves an iterative debiasing procedure that updates initial nuisance function estimates by minimizing a pre-specified loss function along a cleverly chosen submodel. While this approach can be statistically efficient, the iterative nature of TMLE can become computationally expensive in high dimensions, especially when the number of iterations required for convergence is large. Furthermore, the performance of TMLE can be sensitive to the choice of submodel and the initial nuisance function estimates.
In high-dimensional settings with many covariates, the performance of both C-DML and TMLE can be influenced by the curse of dimensionality. The product rate condition, which requires the product of the nuisance function estimation errors to vanish at a rate faster than $n^{-1/2}$, can be challenging to satisfy when the covariate space is high-dimensional. This is because the rates of convergence for many nonparametric estimators deteriorate as the number of covariates increases.
In summary:

C-DML: Can be computationally more efficient than TMLE in high dimensions due to its non-iterative nature. Cross-fitting can further improve finite-sample performance.
TMLE: Can be statistically efficient but computationally more expensive in high dimensions due to its iterative debiasing procedure. Performance can be sensitive to the choice of submodel and initial nuisance function estimates.
The relative performance of C-DML and TMLE in high-dimensional settings is likely to depend on the specific data-generating process, the choice of machine learning algorithms for nuisance function estimation, and the implementation details of each method. Further research and empirical comparisons are needed to provide more definitive guidance on their relative strengths and weaknesses in high-dimensional scenarios.

Could the C-DML framework be extended to handle situations where both the outcome regression and the Riesz representer are estimated inconsistently, potentially by leveraging alternative calibration or debiasing techniques?

The C-DML framework, as presented in the paper, primarily focuses on scenarios where at least one of the nuisance functions is consistently estimated. This is because the DRAL property relies on the ability to express the cross-product remainder term in terms of the estimation error of the most favorable nuisance estimator, which requires at least one of the nuisance functions to be consistently estimated.
However, exploring extensions of C-DML to handle situations where both the outcome regression and the Riesz representer are estimated inconsistently is an interesting avenue for future research. Achieving doubly robust inference in such scenarios would require addressing the bias arising from both nuisance function estimation errors simultaneously.
Potential approaches to extend C-DML for inconsistent nuisance function estimation:

Alternative Calibration Techniques: Exploring alternative calibration techniques that can effectively reduce bias even when the target function is not consistently estimated could be promising. For example, instead of targeting the conditional expectation, one could consider calibration methods that focus on other features of the conditional distribution, such as quantiles or modes.
Joint Debiasing: Developing methods for jointly debiasing both the outcome regression and the Riesz representer estimates could be another direction. This might involve formulating a system of estimating equations that simultaneously target the bias in both nuisance functions.
Instrumental Variable Approaches: In situations where instrumental variables are available, leveraging them to construct doubly robust estimators that are robust to inconsistent estimation of both the outcome regression and the Riesz representer could be explored.
Challenges and Considerations:

Theoretical Justification: Extending C-DML to handle inconsistent nuisance function estimation would require developing new theoretical results that guarantee the validity of the resulting inference. This might involve relaxing the assumption of consistent estimation and deriving new asymptotic properties for the estimators.
Computational Complexity: Jointly debiasing both nuisance functions or employing more sophisticated calibration techniques could increase the computational complexity of the method.
Finite-Sample Performance: The finite-sample performance of any proposed extension would need to be carefully evaluated through simulations and empirical studies.
In conclusion, while the current C-DML framework primarily focuses on scenarios with at least one consistently estimated nuisance function, exploring extensions to handle inconsistent estimation of both nuisance functions is a worthwhile research direction. Leveraging alternative calibration or debiasing techniques could potentially lead to novel doubly robust inference methods with broader applicability.

The paper focuses on linear functionals of the outcome regression. How could the principles of C-DML be applied to develop doubly robust inference methods for other classes of causal parameters, such as those involving non-linear relationships or interactions between treatment and covariates?

While the paper focuses on linear functionals of the outcome regression, the core principles of C-DML, namely calibration and cross-fitting, can potentially be extended to develop doubly robust inference methods for a broader class of causal parameters, including those involving non-linear relationships or interactions.
Here's how C-DML principles could be adapted:

Identify Relevant Nuisance Functions: For non-linear functionals or those involving interactions, the first step is to identify the relevant nuisance functions that need to be estimated. These nuisance functions might include conditional expectations of more complex functions of the outcome, treatment, and covariates. For example, for parameters involving treatment-covariate interactions, the nuisance functions might involve conditional expectations of the product of treatment and covariate values.

Develop Calibration Methods: Once the relevant nuisance functions are identified, the next step is to develop calibration methods that can effectively reduce the bias in their estimates. This might involve extending existing calibration techniques, such as isotonic calibration, to handle more complex functional forms or developing entirely new calibration methods tailored to the specific problem.

Adapt Cross-Fitting: Cross-fitting can still be employed to further reduce bias and improve finite-sample performance. However, the specific implementation of cross-fitting might need to be adapted to the structure of the nuisance functions and the chosen calibration method.

Specific Examples:

Non-linear Outcome Regression: For parameters involving non-linear transformations of the outcome, one could consider estimating the conditional expectation of the transformed outcome as a nuisance function and then calibrating this estimate using techniques that account for the non-linearity.

Treatment-Covariate Interactions: For parameters involving interactions between treatment and covariates, one could estimate the conditional expectation of the interaction term as a nuisance function and then calibrate this estimate using methods that can handle the product structure of the interaction term.
Challenges and Considerations:

Theoretical Challenges: Extending C-DML to non-linear functionals or those involving interactions would require developing new theoretical results that guarantee the validity of the resulting inference. This might involve deriving new influence functions, establishing new asymptotic properties, and potentially imposing additional regularity conditions.

Computational Complexity: Estimating and calibrating more complex nuisance functions can increase the computational complexity of the method.

Choice of Calibration Method: The choice of calibration method will be crucial for the performance of the resulting estimator. Developing effective calibration methods for complex nuisance functions can be challenging.
In conclusion, while extending C-DML to handle non-linear functionals or those involving interactions presents theoretical and practical challenges, the core principles of calibration and cross-fitting provide a promising framework for developing doubly robust inference methods for a wider range of causal parameters. Further research is needed to develop specific methods and establish their theoretical properties and empirical performance.