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.