  It is increasingly common in a wide variety of applied settings to collect
data of multiple different types on the same set of samples. Our particular
focus in this article is on studying relationships between such multiview
features and responses. A motivating application arises in the context of
precision medicine where multi-omics data are collected to correlate with
clinical outcomes. It is of interest to infer dependence within and across
views while combining multimodal information to improve the prediction of
outcomes. The signal-to-noise ratio can vary substantially across views,
motivating more nuanced statistical tools beyond standard late and early
fusion. This challenge comes with the need to preserve interpretability, select
features, and obtain accurate uncertainty quantification. We propose a joint
additive factor regression model (JAFAR) with a structured additive design,
accounting for shared and view-specific components. We ensure identifiability
via a novel dependent cumulative shrinkage process (D-CUSP) prior. We provide
an efficient implementation via a partially collapsed Gibbs sampler and extend
our approach to allow flexible feature and outcome distributions. Prediction of
time-to-labor onset from immunome, metabolome, and proteome data illustrates
performance gains against state-of-the-art competitors. Our open-source
software (R package) is available at https://github.com/niccoloanceschi/jafar.
