  The mass estimator used to calculate black hole (BH) masses in broad-line
active galactic nuclei (AGNs) relies on a virial coefficient (the "$f$ factor")
that is determined by comparing reverberation-mapped (RM) AGNs with measured
bulge stellar velocity dispersions against the $M_{\rm BH}-\sigma_*$ relation
of inactive galaxies. It has recently been recognized that only classical
bulges and ellipticals obey a tight $M_{\rm BH}-\sigma_*$ relation;
pseudobulges have a different zero point and much larger scatter. Motivated by
these developments, we reevaluate the $f$ factor for RM AGNs with available
$\sigma_*$ measurements, updated H$\beta$ RM lags, and new bulge
classifications based on detailed decomposition of high-resolution ground-based
and space-based images. Separate calibrations are provided for the two bulge
types, whose virial coefficients differ by a factor of $\sim 2$: $f=6.3\pm1.5$
for classical bulges and ellipticals and $f = 3.2\pm0.7$ for pseudobulges. The
structure and kinematics of the broad-line region, at least as crudely encoded
in the $f$ factor, seems to related to the large-scale properties or formation
history of the bulge. Lastly, we investigate the bulge stellar masses of the RM
AGNs, show evidence for recent star formation in the AGN hosts that correlates
with Eddington ratio, and discuss the potential utility of the $M_{\rm
BH}-M_{\rm bulge}$ relation as a more promising alternative to the
conventionally used $M_{\rm BH}-\sigma_*$ relation for future refinement of the
virial mass estimator for AGNs.
