  While stellar rotation periods $P_\mathrm{rot}$ may be measured from
broadband photometry, the photometric modulation becomes harder to detect for
slower rotators, which could bias measurements of the long-period tail of the
$P_\mathrm{rot}$ distribution. Alternatively, the $P_\mathrm{rot}$ distribution
of stars can be inferred from their projected rotation velocities $v\sin i$ and
radii $R$, without being biased against photometrically quiet stars. We solve
this inference problem using a hierarchical Bayesian framework, which (i) is
applicable to heteroscedastic measurements of $v\sin i$ and $R$ with
non-Gaussian uncertainties and (ii) does not require a simple parametric form
for the true $P_\mathrm{rot}$ distribution. We test the method on simulated
data sets and show that the true $P_\mathrm{rot}$ distribution can be recovered
from $\gtrsim 100$ sets of $v\sin i$ and $R$ measured with precisions of
$1\,\mathrm{km/s}$ and $4\%$, respectively, unless the true distribution
includes sharp discontinuities. We apply the method to a sample of 144
late-F/early-G dwarfs in the Kepler field with $v\sin i$ measured from
Keck/HIRES spectra, and find that the typical rotation periods of these stars
are similar to the photometric periods measured from Kepler light curves: we do
not find a large population of slow rotators that are missed in the photometric
sample, although we find evidence that the photometric sample is biased for
young, rapidly-rotating stars. Our results also agree with asteroseismic
measurements of $P_\mathrm{rot}$ for Kepler stars with similar ages and
effective temperatures, and show that $\approx 1.1\,M_\odot$ stars beyond the
middle of their main-sequence lifetimes rotate faster than predicted by
standard magnetic braking laws.
