Regarding the issue of sparsity, some exposures are breakdown products of the same compound, so it is unlikely that only one exposure has an effect on the outcome. Also, it is statistically challenging to tell apart highly correlated covariates with limited data. For this reason, it is appealing given the data structure to select blocks of correlated exposures together instead of arbitrarily selecting one chemical in a group.

To address these problems, one possibility is to use a Bayesian approach to inference in order to include prior information to reduce dimensionality while characterizing uncertainty through the posterior distribution. There is an immense literature on Bayesian methods for high-dimensional linear regression, including recent algorithms that can scale up to thousands of predictors [@bondell2012consistent], [@rossell2017nonlocal], [@johndrow2017scalable], [@nishimura2018prior]. In addition some articles have explicitly focused on quadratic regression and interaction detection [@zhang2007bayesian], [@cordell2009detecting], [@mackay2014epistasis]. Bayes variable selection and shrinkage approaches will tend to have problems when predictors are highly correlated; this has motivated a literature on Bayesian latent factor regression [@lucas2006sparse], [@Carvalho2008].

Latent factor regression incorporates shared latent variables in the predictor and response components. This provides dimensionality reduction in modeling of the covariance structure in the predictors and characterizing the impact of 