A Unified Theory for Causal Inference: Direct Debiased Machine Learning via Bregman-Riesz Regression

TL;DR

This paper presents a unified theoretical framework for causal inference that connects various estimation methods like Riesz regression, covariate balancing, DRE, TMLE, and matching, highlighting their relationships and equivalences.

Contribution

It introduces a comprehensive theory unifying multiple causal inference techniques through the concept of Riesz regression and density ratio estimation, clarifying their interconnections.

Findings

Riesz regression is equivalent to density ratio estimation in ATE.

Nearest neighbor matching is a special case of DRE and Riesz regression.

TMLE effectively eliminates leading bias in regression estimators.

Abstract

This note introduces a unified theory for causal inference that integrates Riesz regression, covariate balancing, density-ratio estimation (DRE), targeted maximum likelihood estimation (TMLE), and the matching estimator in average treatment effect (ATE) estimation. In ATE estimation, the balancing weights and the regression functions of the outcome play important roles, where the balancing weights are referred to as the Riesz representer, bias-correction term, and clever covariates, depending on the context. Riesz regression, covariate balancing, DRE, and the matching estimator are methods for estimating the balancing weights, where Riesz regression is essentially equivalent to DRE in the ATE context, the matching estimator is a special case of DRE, and DRE is in a dual relationship with covariate balancing. TMLE is a method for constructing regression function estimators such that the leading bias term becomes zero. Nearest Neighbor Matching is equivalent to Least Squares Density Ratio Estimation and Riesz Regression.