  A hierarchical Bayesian model is applied to the Cosmicflows-3 catalog of
galaxy distances in order to derive the peculiar velocity field and
distribution of matter within $z \sim 0.054$. The model assumes the
$\Lambda$CDM model within the linear regime and includes the fit of the galaxy
distances together with the underlying density field. By forward modeling the
data, the method is able to mitigate biases inherent to peculiar velocity
analyses, such as the Homogeneous Malmquist bias or the log-normal distribution
of peculiar velocities. The statistical uncertainty on the recovered velocity
field is about 150 km/s depending on the location, and we study systematics
coming from the selection function and calibration of distance indicators. The
resulting velocity field and related density fields recover the cosmography of
the Local Universe which is presented in an unprecedented volume of universe 10
times larger than previously reached. This methodology open the doors to
reconstruction of initial conditions for larger and more accurate constrained
cosmological simulations. This work is also preparatory to larger peculiar
velocity datasets coming from Wallaby, TAIPAN or LSST.
