  Using a sample of red giant stars from the Apache Point Observatory Galactic
Evolution Experiment (APOGEE) Data Release 16, we infer the conditional
distribution $p([\alpha/\text{Fe}]\,|\,[\text{Fe/H}])$ in the Milky Way disk
for the $\alpha$-elements Mg, O, Si, S, and Ca. In each bin of [Fe/H] and
Galactocentric radius $R$, we model $p([\alpha/\text{Fe}])$ as a sum of two
Gaussians, representing "low-$\alpha$" and "high-$\alpha$" populations with
scale heights $z_1=0.45\,\text{kpc}$ and $z_2=0.95\,\text{kpc}$, respectively.
By accounting for age-dependent and $z$-dependent selection effects in APOGEE,
we infer the [$\alpha$/Fe] distributions that would be found for a fair sample
of long-lived stars covering all $z$. Near the Solar circle, this distribution
is clearly bimodal at sub-solar [Fe/H], with the low-$\alpha$ and high-$\alpha$
peaks separated by a valley that is $\sim 3$ times lower. In agreement with
previous results, we find that the high-$\alpha$ population is more prominent
at smaller $R$, lower [Fe/H], and larger $|z|$, and that the sequence
separation is smaller for Si and Ca than for Mg, O, and S. We find significant
intrinsic scatter in [$\alpha$/Fe] at fixed [Fe/H] for both the low-$\alpha$
and high-$\alpha$ populations, typically $\sim 0.04$-dex. The means,
dispersions, and relative amplitudes of this two-Gaussian description, and the
dependence of these parameters on $R$, [Fe/H], and $\alpha$-element, provide a
quantitative target for chemical evolution models and a test for hydrodynamic
simulations of disk galaxy formation. We argue that explaining the observed
bimodality will probably require one or more sharp transitions in the disk's
gas accretion, star formation, or outflow history in addition to radial mixing
of stellar populations.
