Direct Bayesian Regression for Distribution-valued Covariates

TL;DR

This paper introduces a Bayesian regression method for scalar outcomes with distribution-valued covariates, directly using repeated measures without density estimation, and demonstrates its theoretical optimality and superior empirical performance.

Contribution

It proposes a novel direct Bayesian regression approach for distribution-valued covariates that avoids density estimation and provides theoretical guarantees.

Findings

Method achieves optimal estimation error bounds.

Performs better than density estimation-based approaches.

Validated through simulations and real data analysis.

Abstract

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.