A robust Bayesian approach for causal inference problems

Causal inference concerns finding the treatment effect on subjects along with causal links between the variables and the outcome. However, the underlying heterogeneity between subjects makes the problem practically unsolvable. Additionally, we often need to find a subset of explanatory variables to understand the treatment effect. Currently, variable selection methods tend to maximise the predictive performance of the underlying model, and unfortunately, under limited data, the predictive performance is hard to assess, leading to harmful consequences. To address these issues, in this paper, we consider a robust Bayesian analysis which accounts for abstention in selecting explanatory variables in the high dimensional regression model. To achieve that, we consider a set of spike and slab priors through prior elicitation to obtain a set of posteriors for both the treatment and outcome model. We are specifically interested in the sensitivity of the treatment effect in high dimensional causal inference as well as identifying confounder variables. However, confounder selection can be deceptive in this setting, especially when a predictor is strongly associated with either the treatment or the outcome. To avoid that we apply a post-hoc selection scheme, attaining a smaller set of confounders as well as separate sets of variables which are only related to treatment or outcome model. Finally, we illustrate our method to show its applicability.