Optimal Bayesian Estimation of a Regression Curve, a Conditional Density and a Conditional Distribution

TL;DR

This paper develops optimal Bayesian estimators for regression curves, conditional densities, and distributions across various data types, emphasizing a general framework that avoids specific priors and leverages posterior predictive distributions.

Contribution

It introduces a unified Bayesian approach for estimating multiple related functions without relying on specific prior distributions, broadening applicability.

Findings

Optimal estimators derived for regression, density, and distribution functions

Framework applicable to continuous, discrete, univariate, multivariate, parametric, and non-parametric cases

Use of posterior predictive distribution as the core of Bayesian estimators

Abstract

In this paper several related estimation problems are addressed from a Bayesian point of view and optimal estimators are obtained for each of them when some natural loss functions are considered. Namely, we are interested in estimating a regression curve. Simultaneously, the estimation problems of a conditional distribution function, or a conditional density, or even the conditional distribution itself, are considered. All these problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, parametric and non-parametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function comonly used in estimating a real function of the unknown parameter. The cornerstone of the mentioned Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate these results.