Lemma 4 ∀ δ > O sufficiently small ∀ d ∈ Rn with ∣∣d∣∣ = δ 3 ε > O indepen-
dent of d and a state i ∈ Ωmax s.t.
L(x* + d, S{) — L(x*,S{) > ε
Proof of lemma 4:. The difference can be expressed as
L(x* + d,si) — L(x*r,si) = VL(x*r,si)d + d'V2L(x)i, si)d + 0(3) (15)
where 0(3) is a third order approximation error. Consider the first order
term: From the optimality of x*r follows that
VL(x*r,si)d ≥ O (16)
for some i ∈ Ωmax. Next, fix such an i and consider the second order term.
Since V2L(xr,si) is normal and positive definite, we have
V2L(x*r,si) = U,iDi Ui
where Ui is unitary and
Di = diag(λi,ι... λi,n)
with λij > O being the eigenvalues of V2L(xtr,, si). Then defining λijmin =
minj χi,j
d'V2L(x*,s)d = d'U'iDi Uid
≥ λi,mind'U'Uid (17)
= λi,mind'd (18)
= λi,minδ (19)
Letting λmin = mini∈Ωmax λijmin it follows from (15), (??), and (19) that
L(xr + d, si) — L(xr,si) ≥ λminδ2 + 0(3)
Choosing δ sufficiently small the third order approximation error can be
made arbitrarily small, e.g. smaller than λm2inδ, then choosing ε = λm2"δ
establishes the claim. ■
Next, normalize the transformed objective of the Bayesian decision maker
(6) as follows
—k. 1 ʌ , , z 4
l(∙c) = - ∑√≈bfe⅛i (20)
i=l
12