Hölder Bounds for Sensitivity Analysis in Causal Reasoning
This work addresses the challenge of causal reasoning under unmeasured confounding for researchers and practitioners, representing an incremental improvement in sensitivity analysis methods.
The paper tackles the problem of estimating treatment effects in causal inference when unobserved confounders exist, by deriving tight bounds on confounding bias using Hölder's inequality and validating them experimentally on synthetic datasets.
We examine interval estimation of the effect of a treatment T on an outcome Y given the existence of an unobserved confounder U. Using Hölder'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.