  With its 12 optical filters, the Javalambre-Photometric Local Universe Survey
(J-PLUS) provides an unprecedented multicolor view of the local Universe. The
third data release (DR3) covers 3,192 deg$^2$ and contains 47.4 million
objects. However, the classification algorithms currently implemented in its
pipeline are deterministic and based solely on the sources morphology. Our goal
is classify the sources identified in the J-PLUS DR3 images into stars,
quasi-stellar objects (QSOs), and galaxies. For this task, we present BANNJOS,
a machine learning pipeline that uses Bayesian neural networks to provide the
probability distribution function (PDF) of the classification. BANNJOS is
trained on photometric, astrometric, and morphological data from J-PLUS DR3,
Gaia DR3, and CatWISE2020, using over 1.2 million objects with spectroscopic
classification from SDSS DR18, LAMOST DR9, DESI EDR, and Gaia DR3. Results are
validated using $1.4 10^5$ objects and cross-checked against theoretical model
predictions. BANNJOS outperforms all previous classifiers in terms of accuracy,
precision, and completeness across the entire magnitude range. It delivers over
95% accuracy for objects brighter than $r = 21.5$ mag, and ~90% accuracy for
those up to $r = 22$ mag, where J-PLUS completeness is < 25%. BANNJOS is also
the first object classifier to provide the full probability distribution
function (PDF) of the classification, enabling precise object selection for
high purity or completeness, and for identifying objects with complex features,
like active galactic nuclei with resolved host galaxies. BANNJOS has
effectively classified J-PLUS sources into around 20 million galaxies, 1
million QSOs, and 26 million stars, with full PDFs for each, which allow for
later refinement of the sample. The upcoming J-PAS survey, with its 56 color
bands, will further enhance BANNJOS's ability to detail each source's nature.
