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.
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