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This study introduces a direct Bayesian regression method for distribution-valued data, bypassing density estimation. The approach improves accuracy, especially with limited repeated measures, offering a novel way to analyze complex covariate relationships.

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • Scalar-on-distribution regression models scalar outcomes using distribution-valued covariates.
  • Traditional methods estimate subject-specific densities from repeated measures before regression.
  • This intermediate density estimation can be inefficient and prone to error, especially with sparse data.

Purpose of the Study:

  • To propose a direct linear scalar-on-distribution regression method.
  • To circumvent the need for intermediate density estimation from repeated measures.
  • To provide theoretical guarantees and explore extensions for practical application.

Main Methods:

  • Directly use repeated measures as covariates in a linear regression framework.
  • Employ a Gaussian process prior on the regression function for Bayesian inference.
  • Achieve closed-form or conjugate Bayesian updates.

Main Results:

  • The proposed method achieves optimal estimation error bounds for the regression function.
  • It demonstrates superior performance compared to methods requiring density estimation, particularly with few repeated measures.
  • The model is invariant to the ordering of repeated measures and extends to non-i.i.d. settings.

Conclusions:

  • Directly modeling distribution-valued covariates using repeated measures is feasible and effective.
  • This approach offers a statistically sound and computationally efficient alternative to traditional methods.
  • The study pioneers theoretical analysis in Bayesian regression with distribution-valued covariates.