Probably approximately correct high-dimensional causal effect estimation given a valid adjustment set

TL;DR

This paper studies covariate adjustment for high-dimensional causal effect estimation from a PAC learning perspective, providing bounds on estimation error, algorithms for discovering adjustment sets, and insights into causal structure recovery.

Contribution

It introduces the concept of an ps-Markov blanket, provides bounds on misspecification error, and develops algorithms with sample complexity guarantees for covariate adjustment in high-dimensional settings.

Findings

PAC bounds on covariate adjustment error exponential in adjustment set size

Algorithm for ps-Markov blanket discovery with sample complexity bounds

Methods to identify smaller adjustment sets beyond ps-Markov blankets

Abstract

Accurate estimates of causal effects play a key role in decision-making across applications such as healthcare, economics, and operations. In the absence of randomized experiments, a common approach to estimating causal effects uses \textit{covariate adjustment}. In this paper, we study covariate adjustment for discrete distributions from the PAC learning perspective, assuming knowledge of a valid adjustment set $\bZ$, which might be high-dimensional. Our first main result PAC-bounds the estimation error of covariate adjustment by a term that is exponential in the size of the adjustment set; it is known that such a dependency is unavoidable even if one only aims to minimize the mean squared error. Motivated by this result, we introduce the notion of an \emph{$\eps$-Markov blanket}, give bounds on the misspecification error of using such a set for covariate adjustment, and provide an algorithm for $\eps$-Markov blanket discovery; our second main result upper bounds the sample complexity of this algorithm. Furthermore, we provide a misspecification error bound and a constraint-based algorithm that allow us to go beyond $\eps$-Markov blankets to even smaller adjustment sets. Our third main result upper bounds the sample complexity of this algorithm, and our final result combines the first three into an overall PAC bound. Altogether, our results highlight that one does not need to perfectly recover causal structure in order to ensure accurate estimates of causal effects.