  This article is motivated by the problem of studying the joint effect of
different chemical exposures on human health outcomes. This is essentially a
nonparametric regression problem, with interest being focused not on a black
box for prediction but instead on selection of main effects and interactions.
For interpretability, we decompose the expected health outcome into a linear
main effect, pairwise interactions, and a non-linear deviation. Our interest is
in model selection for these different components, accounting for uncertainty
and addressing non-identifability between the linear and nonparametric
components of the semiparametric model. We propose a Bayesian approach to
inference, placing variable selection priors on the different components, and
developing a Markov chain Monte Carlo (MCMC) algorithm. A key component of our
approach is the incorporation of a heredity constraint to only include
interactions in the presence of main effects, effectively reducing
dimensionality of the model search. We adapt a projection approach developed in
the spatial statistics literature to enforce identifiability in modeling the
nonparametric component using a Gaussian process. We also employ a dimension
reduction strategy to sample the non-linear random effects that aids the mixing
of the MCMC algorithm. The proposed MixSelect framework is evaluated using a
simulation study, and is illustrated using data from the National Health and
Nutrition Examination Survey (NHANES). Code is available on GitHub.

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