Abstract
Integrative factorization methods for multi-omic data estimate factors explaining biological variation. Factors can be treated as covariates to predict an outcome and the factorization can be used to impute missing values. However, no available methods provide a comprehensive framework for statistical inference and uncertainty quantification for these tasks. A novel framework, Bayesian Simultaneous Factorization (BSF), is proposed to decompose multi-omics variation into joint and individual structures simultaneously within a probabilistic framework. BSF uses conjugate normal priors and 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. BSF is then extended to simultaneously predict a continuous or binary phenotype while estimating latent factors, termed Bayesian Simultaneous Factorization and Prediction (BSFP). BSF and BSFP accommodate concurrent imputation, i.e., imputation during the model-fitting process, and full posterior inference for missing data, including "blockwise" missingness. It is shown via simulation that BSFP is competitive in recovering latent variation structure, and demonstrate the importance of accounting for uncertainty in the estimated factorization within the predictive model. The imputation performance of BSF is examined via simulation under missing-at-random and missing-not-at-random assumptions. Finally, BSFP is used to predict lung function based on the bronchoalveolar lavage metabolome and proteome from a study of HIV-associated obstructive lung disease, revealing multi-omic patterns related to lung function decline and a cluster of patients with obstructive lung disease driven by shared metabolomic and proteomic abundance patterns.