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 $sim 2$: $f=6.3pm1.5$ for classical bulges and ellipticals and $f = 3.2pm0.7$ 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.
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