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We present a Bayesian method to cross-match 5,827,988 high proper motion Gaiasources ($\mu>40 \ mas \ yr^{-1}$) to various photometric surveys: 2MASS,AllWISE, GALEX, RAVE, SDSS and Pan-STARRS. To efficiently associate theseobjects across catalogs, we develop a technique that compares themultidimensional distribution of all sources in the vicinity of each Gaia starto a reference distribution of random field stars obtained by extracting allsources in a region on the sky displaced 2$^\prime$. This offset preserves thelocal field stellar density and magnitude distribution allowing us tocharacterize the frequency of chance alignments. The resulting catalog withBayesian probabilities $>$95% has a marginally higher match rate than currentinternal Gaia DR2 matches for most catalogs. However, a significant improvementis found with Pan-STARRS, where $\sim$99.8% of the sample within the Pan-STARRSfootprint is recovered, as compared to a low $\sim$20.8% in Gaia DR2. Usingthese results, we train a Gaussian Process Regressor to calibrate twophotometric metallicity relationships. For dwarfs of $3500<T_{eff}<5280$ K, weuse metallicity values of 4,378 stars from APOGEE and Hejazi et al. (2020) tocalibrate the relationship, producing results with a $1\sigma$ precision of0.12 dex and few systematic errors. We then indirectly infer the metallicity of4,018 stars with $2850<T_{eff}<3500$ K, that are wide companions of primarieswhose metallicities are estimated with our first regressor, to produce arelationship with a $1\sigma$ precision of 0.21 dex and significant systematicerrors. Additional work is needed to better remove unresolved binaries fromthis sample to reduce these systematic errors.

Bayesian Cross-Matching of High Proper Motion Stars in Gaia DR2 and Photometric Metallicities for $\sim$1.7 million K and M Dwarfs