If 
is known, then the estimator of 
that minimizes the mean square error (data average) is directly , regardless of the data.
More generally, when is not fixed then the mean-squared error is given by
Varying with respect to 
and applying Bayes' theorem (see equation 9.1) gives the result that the estimator is the posterior average of the function being estimated [18], equation 9.9. Thus, assuming that the prior for is known, the posterior average of is the average mean square error optimal estimator. Note that in general other error measures than mean square error may be considered, and these lead to different estimators.