  Various problems in Engineering and Statistics require the computation of the
likelihood ratio function of two probability densities. In classical approaches
the two densities are assumed known or to belong to some known parametric
family. In a data-driven version we replace this requirement with the
availability of data sampled from the densities of interest. For most well
known problems in Detection and Hypothesis testing we develop solutions by
providing neural network based estimates of the likelihood ratio or its
transformations. This task necessitates the definition of proper optimizations
which can be used for the training of the network. The main purpose of this
work is to offer a simple and unified methodology for defining such
optimization problems with guarantees that the solution is indeed the desired
function. Our results are extended to cover estimates for likelihood ratios of
conditional densities and estimates for statistics encountered in local
approaches.
