  The Legacy Survey of Space and Time (LSST) at the Vera C. Rubin Observatory
will discover tens of thousands of extragalactic transients each night. The
high volume of alerts demands immediate classification of transient types in
order to prioritize observational follow-ups before events fade away. We use
host galaxy features to classify transients, thereby providing classification
upon discovery. In contrast to past work that focused on distinguishing Type Ia
and core-collapse supernovae (SNe) using host galaxy features that are not
always accessible (e.g., morphology), we determine the relative likelihood
across $12$ transient classes based on only 19 host apparent magnitudes and
colors from $10$ optical and IR photometric bands. We develop both binary and
multiclass classifiers, using kernel density estimation to estimate the
underlying distribution of host galaxy properties for each transient class.
Even in this pilot study, and ignoring relative differences in transient class
frequencies, we distinguish eight transient classes at purities significantly
above the 8.3% baseline (based on a classifier that assigns labels uniformly
and at random): tidal disruption events ($48\%\pm27\%$, where $\pm$ indicates
the 95% confidence limit), SNe Ia-91bg ($32\%\pm18\%$), SNe Ia-91T
($23\%\pm11\%$), SNe Ib ($23\%\pm13\%$), SNe II ($17\%\pm2\%$), SNe IIn
($17\%\pm6\%$), SNe II P ($16\%\pm4\%$), and SNe Ia ($10\%\pm1\%$). We
demonstrate that our model is applicable to LSST and estimate that our approach
may accurately classify 59% of LSST alerts expected each year for SNe Ia,
Ia-91bg, II, Ibc, SLSN-I, and tidal disruption events. Our code and dataset are
publically available.
