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 disks gas accretion, star formation, or outflow history in addition to radial mixing of stellar populations.
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