On the Relation between Robust and Bayesian Decision Making



Lemma 4 δ > O sufficiently small dRn 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)dO                          (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 =
min
j 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 = miniΩmax 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

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