Derivation of the Prior Distribution in Bayesian Analysis from the Principle of Statistical Equivalence

The distinction between uniform and logarithmic uniform prior distributions is made in terms of the principle of statistical equivalence, consisting of two statistically equivalent experiments, where the variable and parameter of the distribution exchange their roles. The two choices of the prior correspond to two terms in the drift of a diffusion process, and the condition for a stationary solution eliminates the choice of the uniform prior. Parameter randomization gives rise to a new distribution where the parameter of the original distribution is replaced by a 'hitting point' value of the variate.