I show that when the observables (π_E, t_E, θ_E, π_s, µ_s) are well measured up to a discrete degeneracy in the microlensing parallax vector π_E, the relative likelihood of the different solutions can be written in closed form P_i = KH_iB_i, where H_i is the number of stars (potential lenses) having the mass and kinematics of the inferred parameters of solution i and B_i is an additional factor that is formally derived from the Jacobian of the transformation from Galactic to microlensing parameters. Here t_E is the Einstein timescale, θ_E is the angular Einstein radius, and (π_s, µ_s) are the (parallax, proper motion) of the microlensed source. The Jacobian term B_i constitutes an explicit evaluation of the "Rich Argument", i.e., that there is an extra geometric factor disfavoring large-parallax solutions in addition to the reduced frequency of lenses given by H_i. I also discuss how this analytic expression degrades in the presence of finite errors in the measured observables.

We analyze the spatially resolved kinematics of gas and stars for a sample of ten hidden type 1 AGNs in order to investigate the nature of their central sources and the scaling relation with host galaxy stellar velocity dispersion. We select our sample from a large number of hidden type 1 AGNs, which are identified based on the presence of a broad (full width at half maximum ≳1000 km s^-1) component in the Hα line profile and which are frequently mis-classified as type 2 AGNs because AGN continuum and broad emission lines are weak or obscured in the optical spectral range. We used the Blue Channel Spectrograph at the 6.5-m Multiple Mirror Telescope to obtain long-slit data with a spatial scale of 0.3 arcsec pixel^-1. We detected broad Hβ lines for only two targets; however, the presence of strong broad Hα lines indicates that the AGNs we selected are all low-luminosity type 1 AGNs. We measured the velocity, velocity dispersion, and flux of stellar continuum and gas emission lines (i.e., Hβ and [O III]) as a function of distance from the center. The spatially resolved gas kinematics traced by Hβ or [O III] are generally similar to the stellar kinematics except for the inner center, where signatures of gas outflows are detected. We compare the luminosity-weighted effective stellar velocity dispersions with the black hole masses and find that our hidden type 1 AGNs, which have relatively low back hole masses, follow the same scaling relation as reverberation-mapped type 1 AGN and more massive inactive galaxies.