Outcome Assumptions and Duality Theory for Balancing Weights

TL;DR

This paper introduces a new perspective on balancing weight estimators by linking outcome assumptions to convex loss optimization and proposing a quantitative measure for overlap, enhancing robustness in population outcome estimation.

Contribution

It establishes a connection between outcome assumptions and convex loss, replaces the overlap assumption with a minimum worst-case bias measure, and analyzes robustness under assumption misspecification.

Findings

Outcome assumptions lead to convex loss reformulation.

Minimum worst-case bias quantifies overlap more effectively.

Robustness conditions are identified for incorrect outcome assumptions.

Abstract

We study balancing weight estimators, which reweight outcomes from a source population to estimate missing outcomes in a target population. These estimators minimize the worst-case error by making an assumption about the outcome model. In this paper, we show that this outcome assumption has two immediate implications. First, we can replace the minimax optimization problem for balancing weights with a simple convex loss over the assumed outcome function class. Second, we can replace the commonly-made overlap assumption with a more appropriate quantitative measure, the minimum worst-case bias. Finally, we show conditions under which the weights remain robust when our assumptions on the outcomes are wrong.