  We present a Bayesian method to cross-match 5,827,988 high proper motion Gaia
sources ($\mu>40 \ mas \ yr^{-1}$) to various photometric surveys: 2MASS,
AllWISE, GALEX, RAVE, SDSS and Pan-STARRS. To efficiently associate these
objects across catalogs, we develop a technique that compares the
multidimensional distribution of all sources in the vicinity of each Gaia star
to a reference distribution of random field stars obtained by extracting all
sources in a region on the sky displaced 2$^\prime$. This offset preserves the
local field stellar density and magnitude distribution allowing us to
characterize the frequency of chance alignments. The resulting catalog with
Bayesian probabilities $>$95% has a marginally higher match rate than current
internal Gaia DR2 matches for most catalogs. However, a significant improvement
is found with Pan-STARRS, where $\sim$99.8% of the sample within the Pan-STARRS
footprint is recovered, as compared to a low $\sim$20.8% in Gaia DR2. Using
these results, we train a Gaussian Process Regressor to calibrate two
photometric metallicity relationships. For dwarfs of $3500<T_{eff}<5280$ K, we
use metallicity values of 4,378 stars from APOGEE and Hejazi et al. (2020) to
calibrate the relationship, producing results with a $1\sigma$ precision of
0.12 dex and few systematic errors. We then indirectly infer the metallicity of
4,018 stars with $2850<T_{eff}<3500$ K, that are wide companions of primaries
whose metallicities are estimated with our first regressor, to produce a
relationship with a $1\sigma$ precision of 0.21 dex and significant systematic
errors. Additional work is needed to better remove unresolved binaries from
this sample to reduce these systematic errors.
