H\"older Bounds for Sensitivity Analysis in Causal Reasoning

TL;DR

This paper develops tight bounds on the bias caused by unobserved confounders in causal effect estimation using Hölder's inequality, enabling better sensitivity analysis and calibration strategies.

Contribution

It introduces a novel approach to bounding confounding bias in causal inference using Hölder's inequality, with practical calibration methods and validation on synthetic data.

Findings

Derived tight bounds on confounding bias based on unmeasured confounder strength.

Proposed calibration strategies for interval estimation of treatment effects.

Validated bounds through experiments on synthetic and semi-synthetic datasets.

Abstract

We examine interval estimation of the effect of a treatment T on an outcome Y given the existence of an unobserved confounder U. Using H\"older's inequality, we derive a set of bounds on the confounding bias |E[Y|T=t]-E[Y|do(T=t)]| based on the degree of unmeasured confounding (i.e., the strength of the connection U->T, and the strength of U->Y). These bounds are tight either when U is independent of T or when U is independent of Y given T (when there is no unobserved confounding). We focus on a special case of this bound depending on the total variation distance between the distributions p(U) and p(U|T=t), as well as the maximum (over all possible values of U) deviation of the conditional expected outcome E[Y|U=u,T=t] from the average expected outcome E[Y|T=t]. We discuss possible calibration strategies for this bound to get interval estimates for treatment effects, and experimentally validate the bound using synthetic and semi-synthetic datasets.