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Statistica Sinica 33 (2023), 1959-1984


Pedro Luiz Ramos1, Francisco A. Rodrigues2, Eduardo Ramos2,
Dipak K. Dey and Francisco Louzada2

1Pontificia Universidad Católica de Chile,
University of São Paulo, São Carlos and 3University of Connecticut

Abstract: Using objective priors in Bayesian applications has become a common way of analyzing data without using subjective information. Formal rules are usually used to obtain these prior distributions, and the data provide the dominant information in the posterior distribution. However, these priors are typically improper, and may lead to an improper posterior. Here, for a general family of distributions, we show that the objective priors obtained for the parameters either follow a power law distribution, or exhibit asymptotic power law behavior. As a result, we observe that the exponents of the model are between 0.5 and 1. Understanding this behavior allows us to use the exponent of the power law directly to verify whether such priors lead to proper or improper posteriors. The general family of distributions we consider includes essential models such as the exponential, gamma, Weibull, Nakagami-m, half-normal, Rayleigh, Erlang, and Maxwell Boltzmann distributions, among others. In summary, we show that understanding the mechanisms that describe the shape of a prior provides essential information that can be used to understand the properties of posterior distributions.

Key words and phrases: Bayesian inference, objective prior, power-law.

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