  The Kepler Mission has discovered thousands of planets with radii $<4\
R_\oplus$, paving the way for the first statistical studies of the dynamics,
formation, and evolution of these sub-Neptunes and super-Earths. Planetary
masses are an important physical property for these studies, and yet the vast
majority of Kepler planet candidates do not have theirs measured. A key concern
is therefore how to map the measured radii to mass estimates in this
Earth-to-Neptune size range where there are no Solar System analogs. Previous
works have derived deterministic, one-to-one relationships between radius and
mass. However, if these planets span a range of compositions as expected, then
an intrinsic scatter about this relationship must exist in the population. Here
we present the first probabilistic mass-radius relationship (M-R relation)
evaluated within a Bayesian framework, which both quantifies this intrinsic
dispersion and the uncertainties on the M-R relation parameters. We analyze how
the results depend on the radius range of the sample, and on how the masses
were measured. Assuming that the M-R relation can be described as a power law
with a dispersion that is constant and normally distributed, we find that
$M/M_\oplus=2.7(R/R_\oplus)^{1.3}$, a scatter in mass of $1.9\ M_\oplus$, and a
mass constraint to physically plausible densities, is the "best-fit"
probabilistic M-R relation for the sample of RV-measured transiting
sub-Neptunes ($R_{pl}<4\ R_\oplus$). More broadly, this work provides a
framework for further analyses of the M-R relation and its probable
dependencies on period and stellar properties.
