LAUSR

We use density estimation likelihood-free inference, $\mathrm{\ensuremath{\Lambda}}$ cold dark matter simulations of $\ensuremath{\sim}2M$ 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 nonlinear modeling of observational errors that take into account correlations in the data and non-Gaussian distributions. We obtain an ${M}_{200}$ mass estimate ${M}_{\mathrm{MW}+\mathrm{M}31}=4.{6}_{\ensuremath{-}1.8}^{+2.3}\ifmmode\times\else\texttimes\fi{}{10}^{12}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ (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.