A Method for Determining Optimum Systems Using General Bayes Criterion

A method for obtaining an optimum system using general Bayes criterion is developed. This method is applicable to the cases in which a signal to be estimated depends on certain finite-dimensional vector
U
, and where the input is the sum of a function depending on the vector
U
and a normally distributed noise which is independent of the vector, or may be reduced to this form by some nonlinear transform. The method is also applicable to some problems in which
U
is a vector with countably infinite number of components. As a special case the method yields the solution previously given in which the input is a linear function of
U
. The method affords effective determination of optimum systems designed for the detection and estimation of signals in the presence of noise using various practically adequate criteria under rather general conditions. The optimum system given by the method is in general nonlinear, but in some special cases it may be linear. In particular, the optimum system is linear if the signal and the input are linear functions of components of the vector
U
, which is also normally distributed or is an unknown nonrandom vector which may assume any value, and the loss function is any function or functional of the difference between the signal and its estimate (i.e., of the system error). The application of the method to the problem of signal detection and to certain problems of signal estimation are given.