On the Relation between Robust and Bayesian Decision Making

3 Linking Bayesian and Robust Decision Prob-
lems

The objective of this section is to establish a link between the Bayesian and the
robust decision problems described in the previous section. The main idea is to
change the objective function of the Bayesian decision problem in a way that
the Bayesian’s objective will have the same minimum as the robust objective.

Since the Bayesian’s loss function depends on the action x, altering the loss
function is a back-door through which one can cause the Bayesian to behave as
if her priors were changing across actions. In particular, if the Bayesian was to
maximize a transformed loss function T(L(x,s)) with the property that

T(L(x, s)) = L(x, s) ∙ ——-^,-—ɪ                         (5)

Ps

where p_s is the prior probability for state s, then the Bayesian problem would
be identical to the robust decision problem:

min E [T(L(x, s))]

nΩ₁

= min ^ L(x,s_i)^{1 (x,Sl)}p_i
^∈Ω_s ,                  pi

i=l

= min R(x)
x∈Ω_x

Of course, such a transformed ’loss function’ is not a loss function in the strict
sense since it depends on prior probabilities.

Given that direct equivalence between the two problems requires a Bayesian
loss that depends on priors, the strategy is to construct a sequence of trans-
formed loss functions T^k(L(x,s)) for the Bayesian problem with the property
that these transformed loss functions are independent of the prior. At the same
time the solution to

min E [T^k(L(x, s))]                            (6)

■ ω

which is denoted by x*_k should converge to the robust solution x* as к increases
without bound, i.e.

jɪɪɪɪ ι∣^xk -^x* и =°.

k→∞

<<   4   5   6   7   8   9   10   >>

More intriguing information