  The Discrete Source Classifier (DSC) provides probabilistic classification of
sources in Gaia Data Release 3 using a Bayesian framework and a global prior.
The DSC Combmod classifier in GDR3 achieved for the extragalactic classes
(quasars and galaxies) a high completeness of 92%, but a low purity of 22% due
to contamination from the far larger star class. However, these single metrics
mask significant variation in performance with magnitude and sky position.
Furthermore, a better combination of the individual classifiers is possible.
Here we compute two-dimensional representations of the completeness and the
purity as function of Galactic latitude and source brightness, and also exclude
the Magellanic Clouds where stellar contamination significantly reduces the
purity. Reevaluated on a cleaner validation set and without introducing changes
to the published GDR3 DSC probabilities themselves, we achieve for Combmod
average 2D completenesses of 92% and 95% and average 2D purities of 55% and 89%
for the quasar and galaxy classes, respectively. Since the relative proportions
of extragalactic objects to stars in Gaia is expected to vary significantly
with brightness and latitude, we introduce a new prior as a continuous function
of brightness and latitude, and compute new class probabilities. This variable
prior only improves the performance by a few percentage points, mostly at the
faint end. Significant improvement, however, is obtained by a new additive
combination of Specmod and Allosmod. This classifier, Combmod-$\alpha$,
achieves average 2D completenesses of 82% and 93% and average 2D purities of
79% and 93% for the quasar and galaxy classes, respectively, when using the
global prior. Thus, we achieve a significant improvement in purity for a small
loss of completeness. The improvement is most significant for faint quasars
where the purity rises from 20% to 62%.
