  This paper addresses the problem of designing reliable prediction models that
abstain from predictions when faced with uncertain or out-of-distribution
samples - a recently proposed problem known as Selective Classification in the
presence of Out-of-Distribution data (SCOD). We make three key contributions to
SCOD. Firstly, we demonstrate that the optimal SCOD strategy involves a Bayes
classifier for in-distribution (ID) data and a selector represented as a
stochastic linear classifier in a 2D space, using i) the conditional risk of
the ID classifier, and ii) the likelihood ratio of ID and out-of-distribution
(OOD) data as input. This contrasts with suboptimal strategies from current OOD
detection methods and the Softmax Information Retaining Combination (SIRC),
specifically developed for SCOD. Secondly, we establish that in a
distribution-free setting, the SCOD problem is not Probably Approximately
Correct learnable when relying solely on an ID data sample. Third, we introduce
POSCOD, a simple method for learning a plugin estimate of the optimal SCOD
strategy from both an ID data sample and an unlabeled mixture of ID and OOD
data. Our empirical results confirm the theoretical findings and demonstrate
that our proposed method, POSCOD, out performs existing OOD methods in
effectively addressing the SCOD problem.
