# A Gaussian Process Regression Model for Distribution Inputs

## TL;DR

This paper introduces a new family of Gaussian process models for distribution inputs using transportation-based kernels, enabling efficient forecasting of processes indexed by probability distributions.

## Contribution

It develops positive definite kernels based on Wasserstein distances for Gaussian process modeling with distribution inputs, providing a probabilistic framework and forecasting methods.

## Key findings

- New kernels based on Wasserstein distances for distribution inputs
- Probabilistic characterization of Gaussian processes with these kernels
- Efficient forecasting of distribution-indexed Gaussian processes

## Abstract

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling.