  The gravity field of a giant planet is typically our best window into its
interior structure and composition. Through comparison of a model planet's
calculated gravitational potential with the observed potential, inferences can
be made about interior quantities, including possible composition and the
existence of a core. Necessarily, a host of assumptions go into such
calculations, making every inference about a giant planet's structure strongly
model dependent. In this work we present a more general picture by setting
Saturn's gravity field, as measured during the \emph{Cassini} Grand Finale, as
a likelihood function driving a Markov-Chain Monte Carlo exploration of the
possible interior density profiles. The result is a posterior distribution of
the interior structure that is not tied to assumed composition, thermal state,
or material equations of state. Constraints on interior structure derived in
this Bayesian framework are necessarily less informative, but are also less
biased and more general. These empirical and probabilistic constraints on the
density structure are our main data product which we archive for continued
analysis. We find that the outer half of Saturn's radius is relatively well
constrained, and we interpret our findings as suggesting a significant metal
enrichment, in line with atmospheric abundances from remote sensing. As
expected, the inner half of Saturn's radius is less well-constrained by
gravity, but we generally find solutions that include a significant density
enhancement, which can be interpreted as a core, although this core is often
lower in density and larger in radial extent than typically found by standard
models. This is consistent with a dilute core and/or composition gradients.
