# Predictive density estimation under the Wasserstein loss

## TL;DR

This paper studies predictive density estimation using Wasserstein loss, showing that Bayesian predictive densities based on posterior means can outperform equivariant methods in normal models.

## Contribution

It establishes the optimality of plug-in densities with posterior means under Wasserstein loss for location and scale families, and introduces Bayesian predictive densities that dominate existing methods.

## Key findings

- Bayesian predictive densities based on posterior means are optimal under Wasserstein loss.
- Plug-in densities form a complete class for the considered families.
- Bayesian methods can outperform equivariant predictors in normal models.

## Abstract

We investigate predictive density estimation under the $L^2$ Wasserstein loss for location families and location-scale families. We show that plug-in densities form a complete class and that the Bayesian predictive density is given by the plug-in density with the posterior mean of the location and scale parameters. We provide Bayesian predictive densities that dominate the best equivariant one in normal models.