  Precise Gaia measurements of positions, parallaxes, and proper motions
provide an opportunity to calculate 3D positions and 2D velocities (i.e. 5D
phase-space) of Milky Way stars. Where available, spectroscopic radial velocity
(RV) measurements provide full 6D phase-space information, however there are
now and will remain many stars without RV measurements. Without an RV it is not
possible to directly calculate 3D stellar velocities, however one can infer 3D
stellar velocities by marginalizing over the missing RV dimension. In this
paper, we infer the 3D velocities of stars in the Kepler field in Cartesian
Galactocentric coordinates (vx, vy, vz). We directly calculate velocities for
around a quarter of all Kepler targets, using RV measurements available from
the Gaia, LAMOST and APOGEE spectroscopic surveys. Using the velocity
distributions of these stars as our prior, we infer velocities for the
remaining three-quarters of the sample by marginalizing over the RV dimension.
The median uncertainties on our inferred vx, vy, and vz velocities are around
4, 18, and 4 km/s, respectively. We provide 3D velocities for a total of
148,590 stars in the Kepler field. These 3D velocities could enable kinematic
age-dating, Milky Way stellar population studies, and other scientific studies
using the benchmark sample of well-studied Kepler stars. Although the
methodology used here is broadly applicable to targets across the sky, our
prior is specifically constructed from and for the Kepler field. Care should be
taken to use a suitable prior when extending this method to other parts of the
Galaxy.

Astrophysics of Galaxies;Solar and Stellar Astrophysics