  Using a sample of 69,919 red giants from the SDSS-III/APOGEE Data Release 12,
we measure the distribution of stars in the [$\alpha$/Fe] vs. [Fe/H] plane and
the metallicity distribution functions (MDF) across an unprecedented volume of
the Milky Way disk, with radius $3<R<15$ kpc and height $|z|<2$ kpc. Stars in
the inner disk ($R<5$ kpc) lie along a single track in [$\alpha$/Fe] vs.
[Fe/H], starting with $\alpha$-enhanced, metal-poor stars and ending at
[$\alpha$/Fe]$\sim0$ and [Fe/H]$\sim+0.4$. At larger radii we find two distinct
sequences in [$\alpha$/Fe] vs. [Fe/H] space, with a roughly solar-$\alpha$
sequence that spans a decade in metallicity and a high-$\alpha$ sequence that
merges with the low-$\alpha$ sequence at super-solar [Fe/H]. The location of
the high-$\alpha$ sequence is nearly constant across the disk, however there
are very few high-$\alpha$ stars at $R>11$ kpc. The peak of the midplane MDF
shifts to lower metallicity at larger $R$, reflecting the Galactic metallicity
gradient. Most strikingly, the shape of the midplane MDF changes systematically
with radius, with a negatively skewed distribution at $3<R<7$ kpc, to a roughly
Gaussian distribution at the solar annulus, to a positively skewed shape in the
outer Galaxy. For stars with $|z|>1$ kpc or [$\alpha$/Fe]$>0.18$, the MDF shows
little dependence on $R$. The positive skewness of the outer disk MDF may be a
signature of radial migration; we show that blurring of stellar populations by
orbital eccentricities is not enough to explain the reversal of MDF shape but a
simple model of radial migration can do so.
