A priori generalized gamma distribution in Bayesian balance models

The work is devoted to the study of Bayesian balance models, involving the division of the system parameters into two classes: supporting system functioning positive factors and interfering with the functioning negative factors. The balance index, defined as the ratio of the negative factor to the positive factor, is considered. The formulation of the problem, which consists in finding the main probabilistic characteristics (density, distribution function, and moments) of the balance index of factors having a priori generalized gamma distribution with the parameters of the form of one sign, is studied. The results are formulated in terms of the gamma-exponential function. A number of new properties of the latter are given. It is shown that the given statements are easily reformulated for large-scale mixtures of generalized gamma distributions with parameters of the form of different signs. The obtained results can be widely used in models, which describe the processes and phenomena using distributions with a positive unlimited support.