  Contextual bandits are widely-used in the study of learning-based control
policies for finite action spaces. While the problem is well-studied for
bandits with perfectly observed context vectors, little is known about the case
of imperfectly observed contexts. For this setting, existing approaches are
inapplicable and new conceptual and technical frameworks are required. We
present an implementable posterior sampling algorithm for bandits with
imperfect context observations and study its performance for learning optimal
decisions. The provided numerical results relate the performance of the
algorithm to different quantities of interest including the number of arms,
dimensions, observation matrices, posterior rescaling factors, and
signal-to-noise ratios. In general, the proposed algorithm exposes efficiency
in learning from the noisy imperfect observations and taking actions
accordingly. Enlightening understandings the analyses provide as well as
interesting future directions it points to, are discussed as well.
