Prior probability and uncertainty

Consider a stochastic system with output Y whose probability density (pY \mid \Lambda) is a known function of the parameter \Lambda whose true value is unknown. Our aim is to assign a prior probability density b(\Lambda) for \Lambda using all the available knowledge so that we can assess the probabilistic behavior of Y from the corresponding marginal density q(Y) . Usually the range of \Lambda is known. In addition, by considering the distribution of \Lambda in similar systems, we can define the density f(\Lambda) , about which we may have some knowledge such as E(\Lambda _ 1 ^ {2}) \geq 1/2 , etc. We derive an expression for the uncertainty functional \phi(\cdot) involving f(\Lambda) and b(\Lambda) to quantify the discrepancy between the actual behavior of Y and our assessment of its behavior. We pose a two-person zero-sum game with \phi as the payoff function so that b(\Lambda) is chosen by us to minimize \phi , whereas f(\Lambda) is chosen by nature to maximize \phi . We derive an asymptotic expression for the prior density, work out a few examples, and discuss the advantages of this method over others.