  Binary discrimination between well-defined signal and background datasets is
a problem of fundamental importance in particle physics. With detailed event
simulation and the advent of extensive deep learning tools, identification of
the likelihood ratio has typically been reserved as a computational problem.
However, this approach can obscure overtraining or excessive sensitivity to
tuned features of the simulation that may not be well-defined theoretically.
Here, we present the first analysis of binary discrimination for signal and
background distributions for which their likelihood ratio is infrared and
collinear safe, and can therefore be calculated order-by-order in perturbation
theory. We present explicit, general formulas for receiver operator
characteristic curves and the area under it through next-to-leading order. As a
demonstration of this formalism, we apply it to discrimination of
highly-boosted Higgs decays from gluon splitting to bottom quarks. Effects at
next-to-leading order are first sensitive to the flow of color in the jet and
significantly modify discrimination performance at leading-order. In the limit
of infinite boost, these events can be perfectly discriminated because only the
gluon will radiate at finite angles from the bottom quarks, and we find that
large effects persist at energies accessible at the Large Hadron Collider.
Next-to-leading order is therefore required to qualitatively understand results
using machine-learning methods.
