  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 article, we develop selection-aware
Bayesian methods which: (i) 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
$\ell_1$-regularized algorithms; (ii) strike an inevitable tradeoff between the
quality of model selection and inferential power when the same dataset 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 notably tractable updates when
gradient-based 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.
