  The calibration of the tip of the red giant branch (TRGB) in the $I$-band has
a direct role in determinations of the Hubble constant, a subject of recent
interest due to the discrepancy between direct and indirect estimates of its
value. We present a maximum likelihood (ML) method designed to obtain an
independent calibration of the brightness of TRGB using $Gaia$ parallaxes from
the Early Data Release 3 (EDR3) of Milky Way field Giants at high Galactic
latitude. We adopt simple parameterizations for the Milky Way stellar
luminosity function and density law and and optimize the likelihood of the
observed sample as a function of those parameters. Using parameters to
partially constrain the luminosity function from other galaxies similar to the
Milky Way for which high quality TRGB data are available, we find values of the
TRGB magnitude of $ M_I^{TRGB} = -3.91 \pm 0.05 $ (stat) $ \pm 0.09 $ (sys)
mag, where the systematic uncertainty covers the range of shape parameters
found in our Milky Way sample and in reference galaxies. While APASS Data
Release 9 all-sky photometry is insufficient to provide a reliable constraint
on the shape of the Milky Way luminosity function, we estimate that the
photometry from $Gaia$ Data Release 3 (mid-2022) will allow better constraints
on the shape, and lower statistical uncertainties on the tip by a factor of 3.
With expected releases of improved parallax measurements from $Gaia$, the
method of calibrating the TRGB luminosity from field Giants is expected to
reach $\sim$ 0.01 mag uncertainty, which is an important step toward a precise
TRGB-based determination of the Hubble constant.
