  We introduce a Bayesian nonparametric regression model for data with multiway
(tensor) structure, motivated by an application to periodontal disease (PD)
data. Our outcome is the number of diseased sites measured over four different
tooth types for each subject, with subject-specific covariates available as
predictors. The outcomes are not well-characterized by simple parametric
models, so we use a nonparametric approach with a binomial likelihood wherein
the latent probabilities are drawn from a mixture with an arbitrary number of
components, analogous to a Dirichlet Process (DP). We use a flexible probit
stick-breaking formulation for the component weights that allows for covariate
dependence and clustering structure in the outcomes. The parameter space for
this model is large and multiway: patients $\times$ tooth types $\times$
covariates $\times$ components. We reduce its effective dimensionality, and
account for the multiway structure, via low-rank assumptions. We illustrate how
this can improve performance, and simplify interpretation, while still
providing sufficient flexibility. We describe a general and efficient Gibbs
sampling algorithm for posterior computation. The resulting fit to the PD data
outperforms competitors, and is interpretable and well-calibrated. An
interactive visual of the predictive model is available at
http://ericfrazerlock.com/toothdata/ToothDisplay.html , and the code is
available at https://github.com/lockEF/NonparametricMultiway .
