  In many contexts, customized and weighted classification scores are designed
in order to evaluate the goodness of the predictions carried out by neural
networks. However, there exists a discrepancy between the maximization of such
scores and the minimization of the loss function in the training phase. In this
paper, we provide a complete theoretical setting that formalizes weighted
classification metrics and then allows the construction of losses that drive
the model to optimize these metrics of interest. After a detailed theoretical
analysis, we show that our framework includes as particular instances
well-established approaches such as classical cost-sensitive learning, weighted
cross entropy loss functions and value-weighted skill scores.
