Expected Value Minimization in Information Theoretic Multiple Priors Models

Minimization of the expectation
$E_ { \mathbb {P}}(X)$
of a random variable
$X$
over a family
$\Gamma $
of plausible prior distributions
${ \mathbb {P}}$
is addressed, when
$\Gamma $
is a level set of some convex integral functional. As typical cases,
$\Gamma $
may be an
$I$
-divergence ball or some other
$f$
-divergence ball or Bregman distance ball. Regarding localization of the infimum, we show that whether or not the minimum of
$E_ { \mathbb {P}}(X)$
subject to
${ \mathbb {P}}\in \Gamma $
is attained, the densities of the almost minimizing distributions cluster around an explicitly specified function that may have integral less than 1 if the minimum is not attained. If
$\Gamma $
is an
$f$
-divergence ball of radius
$k$
, the minimum is either attained for any choice of
$k$
or it is not attained when
$k$
is less/larger than a critical value. A conjecture is formulated about extending this result beyond
$f$
-divergence balls