مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

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.