  The use of entropy related concepts goes from physics, such as in statistical
mechanics, to evolutionary biology. The Shannon entropy is a measure used to
quantify the amount of information in a system, and its estimation is usually
made under the frequentist approach. In the present paper, we introduce an
fully objective Bayesian analysis to obtain this measure's posterior
distribution. 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
their respective posterior means are finite. An intensive simulation study is
conducted to select the prior that returns better results in terms of bias,
mean square error, and coverage probabilities. The proposed approach is
illustrated in two datasets, where the first one is related to the Achaemenid
dynasty reign period, and the second data describes the time to failure of an
electronic component in the sugarcane harvest machine.
