[Paper Review] Finite Sample Inference for Targeted Learning

Mark van der Laan|arXiv (Cornell University)|Aug 30, 2017

Advanced Causal Inference Techniques20 references3 citations

TL;DR

This paper proposes four finite-sample inference methods for Targeted Maximum Likelihood Estimation (TMLE) using the highly adaptive Lasso (HAL) that leverage the nonparametric bootstrap to estimate the true sampling distribution of the HAL-TMLE, accounting for higher-order remainder terms. The key contribution is a theoretically justified, asymptotically consistent bootstrap method that improves finite-sample coverage over normal-approximation-based inference, especially in high-dimensional or complex models.

ABSTRACT

The Highly-Adaptive-Lasso(HAL)-TMLE is an efficient estimator of a pathwise differentiable parameter in a statistical model that at minimal (and possibly only) assumes that the sectional variation norm of the true nuisance parameters are finite. It relies on an initial estimator (HAL-MLE) of the nuisance parameters by minimizing the empirical risk over the parameter space under the constraint that sectional variation norm is bounded by a constant, where this constant can be selected with cross-validation. In the formulation of the HALMLE this sectional variation norm corresponds with the sum of absolute value of coefficients for an indicator basis. Due to its reliance on machine learning, statistical inference for the TMLE has been based on its normal limit distribution, thereby potentially ignoring a large second order remainder in finite samples. In this article, we present four methods for construction of a finite sample 0.95-confidence interval that use the nonparametric bootstrap to estimate the finite sample distribution of the HAL-TMLE or a conservative distribution dominating the true finite sample distribution. We prove that it consistently estimates the optimal normal limit distribution, while its approximation error is driven by the performance of the bootstrap for a well behaved empirical process. We demonstrate our general inferential methods for 1) nonparametric estimation of the average treatment effect based on observing on each unit a covariate vector, binary treatment, and outcome, and for 2) nonparametric estimation of the integral of the square of the multivariate density of the data distribution.

Motivation & Objective

• To address the limitation of normal-approximation-based inference in TMLE, which can be inaccurate in finite samples due to large higher-order remainder terms.
• To develop finite-sample confidence intervals for HAL-TMLE that are robust to model complexity and high-dimensional data.
• To establish the theoretical validity of nonparametric bootstrap methods for estimating the true sampling distribution of HAL-TMLE under weak regularity conditions.
• To demonstrate that the bootstrap preserves the asymptotic behavior of HAL-MLEs and improves inference accuracy compared to Wald-type intervals.
• To explore the role of the sectional variation norm as a measure of sparsity and its impact on inference performance.

Proposed method

• The paper uses the nonparametric bootstrap to estimate the finite-sample distribution of the HAL-TMLe, treating the cross-validated sectional variation norm bound as fixed.
• Four bootstrap-based inference methods are proposed: two for estimating the true sampling distribution and two conservative variants that dominate the true distribution.
• The method relies on the HAL-MLE to estimate nuisance parameters with bounded sectional variation norm, selected via cross-validation.
• The canonical gradient (influence curve) of the target parameter is used to define the asymptotically linear expansion of the HAL-TMLE.
• The bootstrap is applied to the exact second-order expansion of the HAL-TMLE to directly estimate its sampling distribution, though this sacrifices the substitution estimator property.
• Theoretical consistency of the bootstrap is proven under weak regularity conditions, with approximation error driven by the bootstrap's performance on well-behaved empirical processes.

Experimental results

Research questions

• RQ1 Can the nonparametric bootstrap provide a consistent estimate of the true finite-sample distribution of the HAL-TMLE, even when the higher-order remainder is large?
• RQ2 How does the choice of sectional variation norm bound, selected via cross-validation, affect the finite-sample performance of HAL-TMLE inference?
• RQ3 Does the bootstrap-based inference outperform standard Wald-type confidence intervals that rely on asymptotic normality?
• RQ4 What is the role of the sectional variation norm as a measure of sparsity in enabling robust, adaptive inference for high-dimensional nonparametric models?
• RQ5 Can the bootstrap be extended to sequential or recursive HAL-TMLEs used in longitudinal causal inference?

Key findings

• The nonparametric bootstrap consistently estimates the optimal normal limit distribution of the HAL-TMLE, with approximation error governed by the bootstrap's performance on empirical processes.
• The bootstrap preserves the asymptotic behavior of the HAL-MLEs for nuisance parameters, providing theoretical support for its use in inference.
• The proposed conservative bootstrap methods dominate the true finite-sample distribution, ensuring valid coverage even when the true sectional variation norm is underestimated.
• Finite-sample confidence intervals based on the bootstrap are more accurate than normal-approximation-based intervals, especially in high-dimensional settings with large nuisance parameter spaces.
• The sectional variation norm serves as a powerful, basis-independent measure of sparsity that enables adaptive, robust estimation and inference under weak regularity conditions.
• The results suggest that the indicator basis representation of functions via infinite linear combinations of indicator functions is uniquely suited for defining complexity, enabling efficient MLE and valid bootstrap inference.