Power laws distributions in objective priors

Using objective priors in Bayesian applications has become a common practice to analyze data without subjective information. Formal rules usually 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 improper posterior. Here, for a general family of distributions, we show that the obtained objective priors for the parameters either follow a power-law distribution or have an asymptotic power-law behavior. As a result, we observed that the exponents of the model are between 0.5 and 1. Understanding these behaviors allows us to easily verify if such priors lead to proper or improper posteriors directly from the exponent of the power-law. The general family considered in our study includes essential models such as Exponential, Gamma, Weibull, Nakagamim, Half-Normal, Rayleigh, Erlang, and Maxwell Boltzmann distributions, to list a few. In summary, we show that comprehending the mechanisms describing the shapes of the priors provides essential information that can be used to understand the properties of the posterior distributions.