  Understanding of the pathophysiology of obstructive lung disease (OLD) is
limited by available methods to examine the relationship between multi-omic
molecular phenomena and clinical outcomes. Integrative factorization methods
for multi-omic data can reveal latent patterns of variation describing
important biological signal. However, most methods do not provide a framework
for inference on the estimated factorization, simultaneously predict important
disease phenotypes or clinical outcomes, nor accommodate multiple imputation.
To address these gaps, we propose Bayesian Simultaneous Factorization (BSF). We
use conjugate normal priors and show that the posterior mode of this model can
be estimated by solving a structured nuclear norm-penalized objective that also
achieves rank selection and motivates the choice of hyperparameters. We then
extend BSF to simultaneously predict a continuous or binary response, termed
Bayesian Simultaneous Factorization and Prediction (BSFP). BSF and BSFP
accommodate concurrent imputation and full posterior inference for missing
data, including "blockwise" missingness, and BSFP offers prediction of
unobserved outcomes. We show via simulation that BSFP is competitive in
recovering latent variation structure, as well as the importance of propagating
uncertainty from the estimated factorization to prediction. We also study the
imputation performance of BSF via simulation under missing-at-random and
missing-not-at-random assumptions. Lastly, we use BSFP to predict lung function
based on the bronchoalveolar lavage metabolome and proteome from a study of
HIV-associated OLD. Our analysis reveals a distinct cluster of patients with
OLD driven by shared metabolomic and proteomic expression patterns, as well as
multi-omic patterns related to lung function decline. Software is freely
available at https://github.com/sarahsamorodnitsky/BSFP .
