Prior probability and uncertainty

Consider a stochastic system with output whose probability density is a known function of the parameter whose true value is unknown. Our aim is to assign a prior probability density for using all the available knowledge so that we can assess the probabilistic behavior of from the corresponding marginal density . Usually the range of is known. In addition, by considering the distribution of in similar systems, we can define the density , about which we may have some knowledge such as , etc. We derive an expression for the uncertainty functional involving and to quantify the discrepancy between the actual behavior of and our assessment of its behavior. We pose a two-person zero-sum game with as the payoff function so that is chosen by us to minimize , whereas is chosen by nature to maximize . We derive an asymptotic expression for the prior density, work out a few examples, and discuss the advantages of this method over others.