  Space observatories like the Hubble Space Telescope and Gaia are providing
unprecedented 6D phase space information of satellite galaxies. Such
measurements can shed light on the structure and assembly history of the Local
Group, but improved statistical methods are needed to use them efficiently.
Here we illustrate such a method using analogs of the Local Group's two most
massive satellite galaxies, the Large Magellanic Cloud (LMC) and Triangulum
(M33), from the Illustris dark-matter-only cosmological simulation. We use a
Bayesian inference scheme combining measurements of positions, velocities, and
specific orbital angular momenta ( j ) of the LMC/M33 with importance sampling
of their simulated analogs to compute posterior estimates of the Milky Way (MW)
and Andromeda's (M31) halo masses. We conclude the resulting host halo mass is
more susceptible to bias when using measurements of the current position and
velocity of satellites, especially when satellites are at short-lived phases of
their orbits (i.e. at pericentre). Instead, the j value of a satellite is
well-conserved over time and provides a more reliable constraint on host mass.
The inferred virial mass of the MW (M31) using j of the LMC (M33) is $\rm
M_{vir, MW}=1.02^{+0.77}_{-0.55}\times10^{12}\; M_{\odot}$ ($\rm M_{vir,
M31}=1.37^{+1.39}_{-0.75}\times10^{12}\; M_{\odot}$). Choosing simulated
analogs whose j values are consistent with the conventional picture of a
previous (< 3 Gyr ago), close encounter (< 100 kpc) of M33 about M31 results in
a very low virial mass for M31 ($\rm\sim\!10^{12}\; M_{\odot}$). This supports
the new scenario put forth in Patel et al. (2017), wherein M33 is on its first
passage about M31 or on a long period orbit. We conclude that this Bayesian
inference scheme, utilising satellite j, is a promising method to reduce the
current factor of two spread in the mass range of the MW and M31 moving
forward.
