  Converting a noisy parallax measurement into a posterior belief over distance
requires inference with a prior. Usually this prior represents beliefs about
the stellar density distribution of the Milky Way. However, multi-band
photometry exists for a large fraction of the \textsl{\small{Gaia}}
\textsl{\small{TGAS}} Catalog and is incredibly informative about stellar
distances. Here we use \textsl{\small{2MASS}} colors for 1.4 million
\textsl{\small{TGAS}} stars to build a noise-deconvolved empirical prior
distribution for stars in color--magnitude space. This model contains no
knowledge of stellar astrophysics or the Milky Way, but is precise because it
accurately generates a large number of noisy parallax measurements under an
assumption of stationarity; that is, it is capable of combining the information
from many stars. We use the Extreme Deconvolution (\textsl{\small{XD}})
algorithm---an Empirical Bayes approximation to a full hierarchical model of
the true parallax and photometry of every star---to construct this prior. The
prior is combined with a \textsl{\small{TGAS}} likelihood to infer a precise
photometric parallax estimate and uncertainty (and full posterior) for every
star. Our parallax estimates are more precise than the \textsl{\small{TGAS}}
catalog entries by a median factor of 1.2 (14% are more precise by a factor >2)
and are more precise than previous Bayesian distance estimates that use spatial
priors. We validate our parallax inferences using members of the Milky Way star
cluster M67, which is not visible as a cluster in the \textsl{\small{TGAS}}
parallax estimates, but appears as a cluster in our posterior parallax
estimates. Our results, including a parallax posterior pdf for each of 1.4
million \textsl{\small{TGAS}} stars, are available in companion electronic
tables.
