m = arg min Ey x (y Fm;1(x) ; gm(x)) :
(8)
3 Finite data
The above approach breaks down when the joint distribution of (y x) is represented by a nite
Following the numerical optimization paradigm we take the solution to be
n(xuim)gbN1er.
F (x) = XM fm(x)
m=0
where f0(x) is
de ned by the
oapntiimniitziaatliognuemsse,tahnodd.ffm(x)gM1
to one of parameter optimization
P = arg mPin (P) = arg mPin Ey x (y F (x P)) F (x) = F (x P ):
(3)
For most F (x P) and , numerical optimization methods must be applied tosolve (3). This
expansions of the form
F (x P) = XM mh(x am)
(2)
m=0
wheTrehePfu=ncftimonsamh(gxM0 a. ) in (2) are usually chosen to be simple functions of x with parameters a = fa1 a2 g. Such expansions (2) are at the heart of many function approximation methods
