  $\epsilon$~Eridani is a young planetary system hosting a complex multi-belt
debris disk and a confirmed Jupiter-like planet orbiting at 3.48 AU from its
host star. Its age and architecture are thus reminiscent of the early Solar
System. The most recent study of Mawet et al. 2019, which combined radial
velocity (RV) data and Ms-band direct imaging upper limits, started to
constrain the planet's orbital parameters and mass, but are still affected by
large error bars and degeneracies. Here we make use of the most recent data
compilation from three different techniques to further refine
$\epsilon$~Eridani~b's properties: RVs, absolute astrometry measurements from
the Hipparcos~and Gaia~missions, and new Keck/NIRC2 Ms-band vortex coronagraph
images. We combine this data in a Bayesian framework. We find a new mass, $M_b$
= $0.66_{-0.09}^{+0.12}$~M$_{Jup}$, and inclination, $i$ =
$77.95_{-21.06}^{\circ+28.50}$, with at least a factor 2 improvement over
previous uncertainties. We also report updated constraints on the longitude of
the ascending node, the argument of the periastron, and the time of periastron
passage. With these updated parameters, we can better predict the position of
the planet at any past and future epoch, which can greatly help define the
strategy and planning of future observations and with subsequent data analysis.
In particular, these results can assist the search for a direct detection with
JWST and the Nancy Grace Roman Space Telescope's coronagraph instrument (CGI).
