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

12



More intriguing information

1. Evaluating the Success of the School Commodity Food Program
2. Modellgestützte Politikberatung im Naturschutz: Zur „optimalen“ Flächennutzung in der Agrarlandschaft des Biosphärenreservates „Mittlere Elbe“
3. Langfristige Wachstumsaussichten der ukrainischen Wirtschaft : Potenziale und Barrieren
4. Life is an Adventure! An agent-based reconciliation of narrative and scientific worldviews
5. Neighborhood Effects, Public Housing and Unemployment in France
6. Internationalization of Universities as Internationalization of Bildung
7. The Impact of Financial Openness on Economic Integration: Evidence from the Europe and the Cis
8. Concerns for Equity and the Optimal Co-Payments for Publicly Provided Health Care
9. The name is absent
10. Reform of the EU Sugar Regime: Impacts on Sugar Production in Ireland