Additive Bayesian Variable Selection under Censoring and Misspecification

We discuss the role of misspecification and censoring on Bayesian model selection in the contexts of right-censored survival and concave log-likelihood regression. Misspecification includes wrongly assuming the censoring mechanism to be noninformative. Emphasis is placed on additive accelerated failure time, Cox proportional hazards and probit models. We offer a theoretical treatment that includes local and nonlocal priors, and a general nonlinear effect decomposition to improve power-sparsity trade-offs. We discuss a fundamental question: what solution can one hope to obtain when (inevitably) models are misspecified, and how to interpret it? Asymptotically, covariates that do not have predictive power for neither the outcome nor (for survival data) censoring times, in the sense of reducing a likelihood-associated loss, are discarded. Misspecification and censoring have an asymptotically negligible effect on false positives, but their impact on power is exponential. We show that it can be advantageous to consider simple models that are computationally practical yet attain good power to detect potentially complex effects, including the use of finite-dimensional basis to detect truly nonparametric effects. We also discuss algorithms to capitalize on sufficient statistics and fast likelihood approximations for Gaussian-based survival and binary models.