  Cryogenic scintillating calorimeters are ultrasensitive particle detectors
for rare event searches, particularly for the search for dark matter and the
measurement of neutrino properties. These detectors are made from scintillating
target crystals generating two signals for each particle interaction. The
phonon (heat) signal precisely measures the deposited energy independent of the
type of interacting particle. The scintillation light signal yields particle
discrimination on an event-by-event basis. This paper presents a likelihood
framework modeling backgrounds and a potential dark matter signal in the
two-dimensional plane spanned by phonon and scintillation light energies. We
apply the framework to data from CaWO$_4$-based detectors operated in the
CRESST dark matter search. For the first time, a single likelihood framework is
used in CRESST to model the data and extract results on dark matter in one step
by using a profile likelihood ratio test. Our framework simultaneously fits
(neutron) calibration data and physics (background) data and allows combining
data from multiple detectors. Although tailored to CaWO$_4$-targets and the
CRESST experiment, the framework can easily be expanded to other materials and
experiments using scintillating cryogenic calorimeters for dark matter search
and neutrino physics.
