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