34.9 Exercises 410(a)Ifπis the unique Bayesian optimal policy given priorQ, thenπis admissible.
(b)There is an example whenπis a Bayesian optimal policy andπis inadmissible.
(c) IfEis countable and Supp(Q) =E, thenπis admissible.
34.13(Admissible policies are Bayesian for Bernoulli bandits) LetE
be the set ofk-armed Bernoulli bandits. Prove that every admissible policy is
Bayesian optimal for some prior.Hint Argue that all policies can be written as convex combinations of
deterministic policies using an appropriate linear structure. Then identify the
spaces of environments and policies with compact metric spaces. Let (νj)∞j=1be
a dense subset ofEand repeat the argument in the previous exercise with each
finite subset{ν 1 ,...,νj}and then take the limit asj→∞. You will probably
find Theorem 2.14 useful.
34.14 LetE=EBkbe the space ofk-armed Bernoulli bandits. EndowEwith
a topology via the natural bijection to [0,1]k and letQbe the space of all
probability measures on (E,B(E)) with the weak* topology. Prove that
max
Q∈QBR∗n(Q) =R∗n(E).