  We present an analysis of the fundamental plane of black hole accretion, an
empirical correlation of the mass of a black hole ($M$), its 5 GHz radio
continuum luminosity ($\nu L_{\nu}$), and its 2-10 keV X-ray power-law
continuum luminosity ($L_X$). We compile a sample of black holes with primary,
direct black hole-mass measurements that also have sensitive,
high-spatial-resolution radio and X-ray data. Taking into account a number of
systematic sources of uncertainty and their correlations with the measurements,
we use Markov chain Monte Carlo methods to fit a mass-predictor function of the
form $\log(M/10^{8}\,M_{\scriptscriptstyle \odot}) = \mu_0 + \xi_{\mu R}
\log(L_R / 10^{38}\,\mathrm{erg\,s^{-1}}) + \xi_{\mu X} \log(L_X /
10^{40}\,\mathrm{erg\,s^{-1}})$. Our best-fit results are $\mu_0 = 0.55 \pm
0.22$, $\xi_{\mu R} = 1.09 \pm 0.10$, and $\xi_{\mu X} = -0.59^{+0.16}_{-0.15}$
with the natural logarithm of the Gaussian intrinsic scatter in the log-mass
direction $\ln\epsilon_\mu = -0.04^{+0.14}_{-0.13}$. This result is a
significant improvement over our earlier mass scaling result because of the
increase in active galactic nuclei sample size (from 18 to 30), improvement in
our X-ray binary sample selection, better identification of Seyferts, and
improvements in our analysis that takes into account systematic uncertainties
and correlated uncertainties. Because of these significant improvements, we are
able to consider potential influences on our sample by including all sources
with compact radio and X-ray emission but ultimately conclude that the
fundamental plane can empirically describe all such sources. We end with advice
for how to use this as a tool for estimating black hole masses.
