This work provides a Bayes approach to inference on a particular parameter p[subscript] 1, when the sample likelihood depends on p[subscript] 1 only through a function [theta] = g(p[subscript]1,p[subscript]2) of p[subscript] 1 and a second parameter p[subscript] 2; in this case, the posterior mean of p[subscript] 1 turns out to be a generalized posterior moment of [theta]. An example of this is the case when only system performance data are available, but component performance evaluation is desired;Bayes inference on p[subscript] 1 is addressed from both the small-sample and asymptotic point of view, including the comparison of the posterior mean of p[subscript] 1 with several of its approximations. For a certain special case of the system example above, hypergeometric forms are given for the posterior mean of p[subscript] 1, and its approximations.