  We mine the Tycho-{\it Gaia} astrometric solution (TGAS) catalog for wide
stellar binaries by matching positions, proper motions, and astrometric
parallaxes. We separate genuine binaries from unassociated stellar pairs
through a Bayesian formulation that includes correlated uncertainties in the
proper motions and parallaxes. Rather than relying on assumptions about the
structure of the Galaxy, we calculate Bayesian priors and likelihoods based on
the nature of Keplerian orbits and the TGAS catalog itself. We calibrate our
method using radial velocity measurements and obtain 6196 high-confidence
candidate wide binaries with projected separations $s\lesssim1$ pc. The
normalization of this distribution suggests that at least 0.6\% of TGAS stars
have an associated, distant TGAS companion in a wide binary. We demonstrate
that {\it Gaia}'s astrometry is precise enough that it can detect projected
orbital velocities in wide binaries with orbital periods as large as 10$^6$ yr.
For pairs with $s\ \lesssim\ 4\times10^4$~AU, characterization of random
alignments indicate our contamination to be $\approx$5\%. For $s \lesssim
5\times10^3$~AU, our distribution is consistent with \"{O}pik's Law. At larger
separations, the distribution is steeper and consistent with a power-law
$P(s)\propto s^{-1.6}$; there is no evidence in our data of any bimodality in
this distribution for $s \lesssim$ 1 pc. Using radial velocities, we
demonstrate that at large separations, i.e., of order $s \sim$ 1 pc and beyond,
any potential sample of genuine wide binaries in TGAS cannot be easily
distinguished from ionized former wide binaries, moving groups, or
contamination from randomly aligned stars.
