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Using CaWO$_4$ crystals as cryogenic calorimeters, the CRESST experimentsearches for nuclear recoils caused by the scattering of potential Dark Matterparticles. A reliable identification of a potential signal crucially depends onan accurate background model. In this work we introduce an improvednormalisation method for CRESST's model of the electromagnetic backgrounds.Spectral templates, based on Geant4 simulations, are normalised via a Bayesianlikelihood fit to experimental background data. Contrary to our previous work,no assumption of partial secular equilibrium is required, which results in amore robust and versatile applicability. Furthermore, considering thecorrelation between all background components allows us to explain 82.7% of theexperimental background within [1 keV, 40 keV], an improvement of 18.6%compared to our previous method.

High-Dimensional Bayesian Likelihood Normalisation for CRESST's Background Model