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.

翻译：本文提出了一种完全客观的贝叶斯分析方法，用于获取熵测度的后验分布。我们特别关注伽马分布——该分布描述了物理学、工程学和生物学中的许多自然现象。我们以熵为参数重新参数化模型，并推导了不同的客观先验分布，包括杰弗里斯先验、参考先验和匹配先验。由于所得先验分布是非正则的，我们证明了所得后验分布是正则的，且其各自的后验均值有限。通过密集的模拟研究，筛选出在偏差、均方误差和覆盖概率方面表现更优的先验分布。所提方法在两个数据集上得到验证：第一个数据集涉及阿契美尼德王朝统治时期，第二个数据集描述了甘蔗收割机中电子元件的失效时间。