  This is the eighth in a series of papers reporting on a large reverberation
mapping campaign to measure black hole (BH) mass in high accretion rate active
galactic nuclei (AGNs). We employ the recently developed dynamical modeling
approach for broad-line regions (BLRs) based on the method of Pancoast et al.
to analyze the reverberation mapping dataset of Mrk 142 observed in the first
monitoring season. In this approach, continuum variations are reconstructed
using a damped random walk process, and BLR structure is delineated using a
flexible disk-like geometry, in which BLR clouds move around the central BH
with Keplerian orbits or inflow/outflow motion. The approach also includes the
possibilities of anisotropic emission of BLR clouds, non-linear response of the
line emission to the continuum, and different long-term trends in the continuum
and emission-line variations. We implement the approach in a Bayesian framework
that is apt for parallel computation and use a Markov Chain Monte Carlo
technique to recover the parameters and uncertainties for the modeling,
including mass of the central BH. We apply three BLR models with different
prescriptions of BLR clouds distributions and find that the best model for
fitting the data of Mrk 142 is a two-zone BLR model, consistent with the
theoretical BLR model surrounding slim accretion disks. The best model yields a
BH mass of $\log (M_\bullet/M_\odot)=6.23_{-0.45}^{+0.26}$, resulting in a
virial factor of $\log f=-0.36_{-0.54}^{+0.33}$ for the full width at half
maximum of the H$\beta$ line measured from the mean spectrum. The virial
factors for the other measures of the H$\beta$ line width are also presented.
