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We consider the problem of performing inference for a continuous treatment effect on a binary outcome variable while controlling for high dimensional baseline covariates. We propose a novel Bayesian framework for performing inference for the desired low-dimensional parameter in a high-dimensional logistic regression model. While it is relatively easier to address this problem in linear regression, the nonlinearity of the logistic regression poses additional challenges that make it difficult to orthogonalize the effect of the treatment variable from the nuisance variables. Our proposed approach provides the first Bayesian alternative to the recent frequentist developments and can incorporate available prior information on the parameters of interest, which plays a crucial role in practical applications. In addition, the proposed approach incorporates uncertainty in orthogonalization in high dimensions instead of relying on a single instance of orthogonalization as done by frequentist methods. We provide uniform convergence results that show the validity of credible intervals resulting from the posterior. Our method has competitive empirical performance when compared with state-of-the-art methods. 

Subgroup analysis has emerged as an important tool to identify unknown subgroup memberships in the presence of heterogeneity. However, much of the existing work focused on the low-dimensional scenario where only a few candidate variables are considered for modeling the subgroup membership. In this paper, we propose a two-component structured mixture model with a Bayesian variable selection approach for identifying predictive and prognostic variables separately in the high-dimensional setting. By employing spike and slab priors, we achieve the selection of predictive and prognostic variables and the estimation of the treatment effect in the selected subgroup simultaneously. We establish theoretical properties by showing strong variable selection consistency and posterior contraction behavior of our method, and demonstrate its performance using simulation studies. Finally, we apply the proposed method to data from the National Supported Work and the AIDS Clinical Trials Group 320 study, identifying predictive and prognostic variables associated with subgroups exhibiting differential treatment effects.