On the Relation between Robust and Bayesian Decision Making

Lemma 4 ∀ δ > O sufficiently small ∀ d ∈ Rⁿ with ∣∣d∣∣ = δ 3 ε > O indepen-
dent of d and a state i ∈ Ω_max s.t.

L(x* + d, S{) — L(x*,S{) > ε

Proof of lemma 4:. The difference can be expressed as

L(x* + d,s_i) — L(x*_r,s_i) = VL(x*_r,s_i)d + d'V²L(x)i, s_i)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,s_i)d ≥ O                          (16)

for some i ∈ Ω_max. 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^,_iD_i Ui

where U_i is unitary and

Di = diag(λ_i,ι... λi,n)

with λ_ij > O being the eigenvalues of V²L(x^t_r,, s_i). Then defining λ_ijmin =
minj ^χi,j

d'V²L(x*,s)d = d'U'_iD_i U_id

≥ λi,mind'U'Uid                  (17)

= λi,mind'd                         (18)

= λi,minδ                         (19)

Letting λmin = mini∈Ω_max λi_jmin 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 ^λm₂^inδ, 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

<<   10   11   12   13   14   15   16   >>

More intriguing information