  The likelihood ratio is a crucial quantity for statistical inference in
science that enables hypothesis testing, construction of confidence intervals,
reweighting of distributions, and more. Many modern scientific applications,
however, make use of data- or simulation-driven models for which computing the
likelihood ratio can be very difficult or even impossible. By applying the
so-called ``likelihood ratio trick,'' approximations of the likelihood ratio
may be computed using clever parametrizations of neural network-based
classifiers. A number of different neural network setups can be defined to
satisfy this procedure, each with varying performance in approximating the
likelihood ratio when using finite training data. We present a series of
empirical studies detailing the performance of several common loss functionals
and parametrizations of the classifier output in approximating the likelihood
ratio of two univariate and multivariate Gaussian distributions as well as
simulated high-energy particle physics datasets.
