Bayesian balance models

A number of previous author’s works were devoted to the Bayesian approach to queuing and reliability. In this paper, the method application is extended to a wide range of problems, such as demography, physics, political science, modeling of emergencies, medicine, etc. The method is based on separation of system factors into two groups: those that support functioning of the system (positive, or p-factors) and those that inhibit system’s functioning (negative, or n-factors). In the paper, system’s balance index, which equals to the ratio of n- and p-factors, and the advantage index, which equals to the ratio of p-factor to the sum of n- and p-factors, are considered. It is assumed that the factors, which affect the system, change over time, and besides their exact values are impossible to determine due to the measuring equipment’s imperfections, excessively high expenses on thorough research, lack of time and resources, and so on. Such prerequisites lead to usage of the Bayesian method in application to the problems described. The method implies randomization of the initial parameters (factors) and, as a consequence, randomization of the balance and advantage indices. The main goal of the research is to study probabilistic characteristics of the balance and advantage indices assuming that the apriori distributions of the system’s factors are known. In the case of independently distributed n- and p-factors, which are random variables, the problem is reduced to studying properties of the distributions’ mixtures. As opposed to popular normal mixtures, in Bayesian balance models, the distribution being mixed has a positive support. Special attention is paid to apriori gamma-type distributions, since these distributions are adequate asymptotic approximations of a wide range of probability distributions. The mixtures of exponential, Erlang, and Weibull apriori distributions were considered earlier. In this paper, special attention is paid to the case of Nakagami m-distribution of n- and p-factors (with its particular cases of Rayleigh, Maxwell–Boltzmann, chi-, and other distributions). The explicit formulas for density, distribution functions, and moments of the balance index for different combinations of distributions are obtained. The results provided in this paper can be applied to many different tasks conserning indices, ratings, and indicators.