  In this manuscript, we study the problem of scalar-on-distribution
regression; that is, instances where subject-specific distributions or
densities, or in practice, repeated measures from those distributions, are the
covariates related to a scalar outcome via a regression model. We propose a
direct regression for such distribution-valued covariates that circumvents
estimating subject-specific densities and directly uses the observed repeated
measures as covariates. The model is invariant to any transformation or
ordering of the repeated measures. Endowing the regression function with a
Gaussian Process prior, we obtain closed form or conjugate Bayesian inference.
Our method subsumes the standard Bayesian non-parametric regression using
Gaussian Processes as a special case. Theoretically, we show that the method
can achieve an optimal estimation error bound. To our knowledge, this is the
first theoretical study on Bayesian regression using distribution-valued
covariates. Through simulation studies and analysis of activity count dataset,
we demonstrate that our method performs better than approaches that require an
intermediate density estimation step.
