  We have extended the compressible liquid-drop model (CLDM) with a
density-dependent surface term (eCLDM), which allows for a unified description
of both the nuclear ground state energies and the incompressibility modulus in
finite nuclei $K_A$. We analyse the role of the nuclear empirical parameters,
e.g., $K_{sat}$, $Q_{sat}$, $L_{sym}$ and $K_{sym}$, which contribute to the
bulk properties, as well as the role of the finite size contributions. For the
bulk properties, the density and isospin dependencies of the nuclear
incompressibility in infinite matter are characterized by introducing new
empirical parameters, and two new constraints for the value of $K_{sym}$ are
suggested. For finite nuclei, we employ a Bayesian approach coupled to a
Markov-Chain Monte-Carlo (MCMC) exploration of the parameter space to confront
the model predictions of $K_A$ in Zr, Sn and Pb isotopes to the experimental
data. We show that $Q_{sat}\approx -950\pm 200$~MeV describes the experimental
measurements of $K_A$ in these isotopes. This value is different from the ones
deduced from phenomenological nuclear energy density functionals, suggesting a
possible explanation of their difficulty to accurately describe Zr, Sn and Pb
data all together. In addition we explore the impact of a fictitious
measurement of the Giant Monopole Resonance energy in $^{132}$Sn. We show that
this measurement, provided it is accurate enough, will allow to better
determine $K_{sym}$ and $K_{\tau}$. Finally we explore the properties of the
sound speed around saturation density and show the important role of finite
size terms in finite nuclei since they reduce the sound speed to approximately
half compared to nuclear matter.
