  This work proposes a new loss function targeting classification problems,
utilizing a source of information overlooked by cross entropy loss. First, we
derive a series of the tightest upper and lower bounds for the probability of a
random variable in a given interval. Second, a lower bound is proposed for the
probability of a true positive for a parametric classification problem, where
the form of probability density function (pdf) of data is given. A closed form
for finding the optimal function of unknowns is derived to maximize the
probability of true positives. Finally, for the case that the pdf of data is
unknown, we apply the proposed boundaries to find the lower bound of the
probability of true positives and upper bound of the probability of false
positives and optimize them using a loss function which is given by combining
the boundaries. We demonstrate that the resultant loss function is a function
of the signal to noise ratio both within and across logits. We empirically
evaluate our proposals to show their benefit for classification problems.
