  We use Density Estimation Likelihood-Free Inference, $\Lambda$ Cold Dark
Matter simulations of $\sim 2M$ galaxy pairs, and data from Gaia and the Hubble
Space Telescope to infer the sum of the masses of the Milky Way and Andromeda
(M31) galaxies, the two main components of the Local Group. This method
overcomes most of the approximations of the traditional timing argument, makes
the writing of a theoretical likelihood unnecessary, and allows the non-linear
modelling of observational errors that take into account correlations in the
data and non-Gaussian distributions. We obtain an $M_{200}$ mass estimate
$M_{\rm MW+M31} = 4.6^{+2.3}_{-1.8} \times 10^{12} M_{\odot}$ ($68 \%$ C.L.),
in agreement with previous estimates both for the sum of the two masses and for
the individual masses. This result is not only one of the most reliable
estimates of the sum of the two masses to date, but is also an illustration of
likelihood-free inference in a problem with only one parameter and only three
data points.
