correlated groups of predictors on the response. Such approaches are closely related to principal components regression, but it tends to be easier to simultaneously incorporate shrinkage and uncertainty quantification within the Bayesian framework. In addition, within the Bayes latent factor regression paradigm, typical identifiability constraints such as orthogonality are not needed (see, for example [@Bhattacharya2013]). The main contribution of this article is to generalize Bayesian latent factor regression to accommodate interactions using an approach inspired by [@wang2019penalized]. This is accomplished by including pairwise interactions in the latent variables in the response component. We refer to the resulting framework as Factor analysis for INteractions (FIN). There is a rich literature on quadratic and nonlinear latent variable modeling, largely in psychometrics (refer, for example, to [@arminger1998bayesian]). However, to our knowledge, such approaches have not been used for inferences on interactions in regression problems.

In *Section 2* we describe the proposed FIN framework, including extensions for higher order interactions. In *Section 3* we provide theory on model misspecification and consistency. *Section 4* contains a simulation study. *Section 5* illustrates the methods on NHANES data. Code is available at <https://github.com/fedfer/factor_interactions>. Proofs of *Proposition 2* and *Proposition 3* are included in the *Supplementary Material*.

Model
