  In the context of finite mixture models one considers the problem of
classifying as many observations as possible in the classes of interest while
controlling the classification error rate in these same classes. Similar to
what is done in the framework of statistical test theory, different type I and
type II-like classification error rates can be defined, along with their
associated optimal rules, where optimality is defined as minimizing type II
error rate while controlling type I error rate at some nominal level. It is
first shown that finding an optimal classification rule boils down to searching
an optimal region in the observation space where to apply the classical Maximum
A Posteriori (MAP) rule. Depending on the misclassification rate to be
controlled, the shape of the optimal region is provided, along with a heuristic
to compute the optimal classification rule in practice. In particular, a
multiclass FDR-like optimal rule is defined and compared to the thresholded MAP
rules that is used in most applications. It is shown on both simulated and real
datasets that the FDR-like optimal rule may be significantly less conservative
than the thresholded MAP rule.
