A NEURAL MEAN EMBEDDING APPROACH FOR BACK-DOOR AND FRONT-DOOR ADJUSTMENT

Abstract

We consider the estimation of average and counterfactual treatment effects, under two settings: back-door adjustment and front-door adjustment. The goal in both cases is to recover the treatment effect without having an access to a hidden confounder. This objective is attained by first estimating the conditional mean of the desired outcome variable given relevant covariates (the "first stage" regression), and then taking the (conditional) expectation of this function as a "second stage" procedure. We propose to compute these conditional expectations directly using a regression function to the learned input features of the first stage, thus avoiding the need for sampling or density estimation. All functions and features (and in particular, the output features in the second stage) are neural networks learned adaptively from data, with the sole requirement that the final layer of the first stage should be linear. The proposed method is shown to converge to the true causal parameter, and outperforms the recent state-of-the-art methods on challenging causal benchmarks, including settings involving high-dimensional image data.

1. INTRODUCTION

The goal of causal inference from observational data is to predict the effect of our actions, or treatments, on the outcome without performing interventions. Questions of interest can include what is the effect of smoking on life expectancy? or counterfactual questions, such as given the observed health outcome for a smoker, how long would they have lived had they quit smoking? Answering these questions becomes challenging when a confounder exists, which affects both treatment and the outcome, and causes bias in the estimation. Causal estimation requires us to correct for this confounding bias. A popular assumption in causal inference is the no unmeasured confounder requirement, which means that we observe all the confounders that cause the bias in the estimation. Although a number of causal inference methods are proposed under this assumption (Hill, 2011; Shalit et al., 2017; Shi et al., 2019; Schwab et al., 2020) , it rarely holds in practice. In the smoking example, the confounder can be one's genetic characteristics or social status, which are difficult to measure for both technical and ethical reasons. To address this issue, Pearl (1995) proposed back-door adjustment and front-door adjustment, which recover the causal effect in the presence of hidden confounders using a back-door variable or frontdoor variable, respectively. The back-door variable is a covariate that blocks all causal effects directed from the confounder to the treatment. In health care, patients may have underlying predispositions to illness due to genetic or social factors (hidden), which cause measurable symptoms. The symptoms can be used as the back-door variable if the treatment is chosen based on these. By contrast, a front-door variable blocks the path from treatment to outcome. In perhaps the bestknown example, the amount of tar in a smoker's lungs serves as a front-door variable, since it is increased by smoking, shortens life expectancy, and has no direct link to underlying (hidden) sociological traits. Pearl (1995) showed that causal quantities can be obtained by taking the (conditional) expectation of the conditional average outcome. While Pearl (1995) only considered the discrete case, this framework was extended to the continuous case by Singh et al. (2020) , using two-stage regression (a review of this and other recent approaches 1

