  In the Milky Way, the distribution of stars in the $[\alpha/\mathrm{Fe}]$ vs.
$[\mathrm{Fe/H}]$ and $[\mathrm{Fe/H}]$ vs. age planes holds essential
information about the history of star formation, accretion, and dynamical
evolution of the Galactic disk. We investigate these planes by applying novel
statistical methods called copulas and elicitable maps to the ages and
abundances of red giants in the APOGEE survey. We find that the low- and
high-$\alpha$ disk stars have a clean separation in copula space and use this
to provide an automated separation of the $\alpha$ sequences using a purely
statistical approach. This separation reveals that the high-$\alpha$ disk ends
at the same [$\alpha$/Fe] and age at high $[\mathrm{Fe/H}]$ as the
low-$[\mathrm{Fe/H}]$ start of the low-$\alpha$ disk, thus supporting a
sequential formation scenario for the high- and low-$\alpha$ disks. We then
combine copulas with elicitable maps to precisely obtain the correlation
between stellar age $\tau$ and metallicity $[\mathrm{Fe/H}]$ conditional on
Galactocentric radius $R$ and height $z$ in the range $0 < R < 20$ kpc and $|z|
< 2$ kpc. The resulting trends in the age-metallicity correlation with radius,
height, and [$\alpha$/Fe] demonstrate a $\approx 0$ correlation wherever
kinematically-cold orbits dominate, while the naively-expected negative
correlation is present where kinematically-hot orbits dominate. This is
consistent with the effects of spiral-driven radial migration, which must be
strong enough to completely flatten the age-metallicity structure of the
low-$\alpha$ disk.
