Objective Bayesian Analysis for the Differential Entropy of the Gamma Distribution

TL;DR

This paper develops an objective Bayesian framework for estimating the differential entropy of the gamma distribution, ensuring proper posteriors and evaluating priors through simulations and real data applications.

Contribution

It introduces a reparametrization of the gamma model in terms of entropy and derives various objective priors, analyzing their properties and performance.

Findings

Proper posterior distributions are obtained despite improper priors.

Simulation results identify the best prior in terms of bias and coverage.

Applications demonstrate the method's effectiveness on real datasets.

Abstract

The present paper introduces a fully objective Bayesian analysis to obtain the posterior distribution of an entropy measure. Notably, we consider the gamma distribution, which describes many natural phenomena in physics, engineering, and biology. We reparametrize the model in terms of entropy, and different objective priors are derived, such as Jeffreys prior, reference prior, and matching priors. Since the obtained priors are improper, we prove that the obtained posterior distributions are proper and that their respective posterior means are finite. An intensive simulation study is conducted to select the prior that returns better results regarding bias, mean square error, and coverage probabilities. The proposed approach is illustrated in two datasets: the first relates to the Achaemenid dynasty reign period, and the second describes the time to failure of an electronic component in a sugarcane harvest machine.