Integrative Bayesian models using Post‐selective inference: A case study in radiogenomics

Integrative analyses based on statistically relevant associations between genomics and a wealth of intermediary phenotypes (such as imaging) provide vital insights into their clinical relevance in terms of the disease mechanisms. Estimates for uncertainty in the resulting integrative models are however unreliable unless inference accounts for the selection of these associations with accuracy. In this paper, we develop selection‐aware Bayesian methods, which (1) counteract the impact of model selection bias through a “selection‐aware posterior” in a flexible class of integrative Bayesian models post a selection of promising variables via ℓ1‐regularized algorithms; (2) strike an inevitable trade‐off between the quality of model selection and inferential power when the same data set is used for both selection and uncertainty estimation. Central to our methodological development, a carefully constructed conditional likelihood function deployed with a reparameterization mapping provides tractable updates when gradient‐based Markov chain Monte Carlo (MCMC) sampling is used for estimating uncertainties from the selection‐aware posterior. Applying our methods to a radiogenomic analysis, we successfully recover several important gene pathways and estimate uncertainties for their associations with patient survival times.