  Actively accreting supermassive black holes significantly impact the
evolution of their host galaxies, truncating further star formation by
expelling large fractions of gas with wide-angle outflows. The X-ray band is
key to understanding how these black hole winds affect their environment, as
the outflows have high temperatures ($\sim$10$^{5-8}$K). We have developed a
Bayesian framework for characterizing Active Galactic Nuclei (AGN) outflows
with an improved ability to explore parameter space and perform robust model
selection. We applied this framework to a new 700 ks and an archival 315 ks
Chandra High Energy Transmission Gratings observation of the Seyfert galaxy NGC
4051. We have detected six absorbers intrinsic to NGC 4051. These wind
components span velocities from 400 km s$^{-1}$ to 30,000 km s$^{-1}$. We have
determined that the most statistically significant wind component is purely
collisionally ionized, which is the first detection of such an absorber. This
wind has $T\approx10^7$ K and $v\approx880$ km s$^{-1}$ and remains remarkably
stable between the two epochs. Other slow components also remain stable across
time. Fast outflow components change their properties between 2008 and 2016,
suggesting either physical changes or clouds moving in and out of the line of
sight. For one of the fast components we obtain one of the tightest wind
density measurements to date, log $n/$[cm$^{-3}$]=13.0$^{+0.01}_{-0.02}$, and
determine that it is located at $\sim$240 gravitational radii. The estimated
total outflow power surpasses 5% of the bolometric luminosity (albeit with
large uncertainties) making it important in the context of galaxy-black hole
interactions.
