An Empirical Bayes Approach to Statistics

Let X be a random variable which for simplicity we shall assume to have discrete values x and which has a probability distribution depending in a known way on an unknown real parameter A, (1) p (xIX) =Pr [X = xIA =X], A-itself being a random variable with a priori distribution function (2) G (X) =Pr [A-< X. The unconditional probability distribution of X is then given by (3) PG(x) =Pr[X=xI =fP(xjX)dG(X), and the expected squared deviation of any estimator of A of the form sp(X) is (4) Ek[o(X) -A2 =E{E[(op(X) -A)21A=X] I =flP(x X)[ (x) -X]2dG(X) ='I (x X) [z (x)] 2dG (X) which is a minimum when so(x) is defined for each x as that value y = y(x) for which (5) I (x) =fp (x I X) (y-X) 2dG (X) = minimum. But for any fixed x the quantity (6) I (x) =y2fpdG-2yfpXdG+fPX2dG fbpdG (Y fpXdG\) + [fp XrCIdG.(fpXdG) 2 Pd-G J fpdG J is a minimum with respect to y when (7) y =rfpdG' the minimum value of I(x) being (8) I a(x) =fp (x I X)2dG (X)[fpp(x I X) dG (X) ]