We have inverted the most recent set of geophysical observations pertinent to lunar interior properties, including mass M and moment of inertia I as determined by Lunar Prospector, the two Love numbers k2 and h2 as well as monthly tidal dissipation Q obtained from more than 35 years of lunar laser ranging data. In wanting to assess the ability of these parameters to constrain lunar mantle and core structure, we have used a stochastically based sampling algorithm to invert the geophysical data to obtain radial density and shear wave velocity profiles. The results indicate a small liquid Fe core (r 1, thereby favouring the fluid core hypothesis. In addition, shear wave velocities for the lower mantle region (depths > 1100 km) are generally found to be lower than upper mantle velocities and can be interpreted as implying the presence of partial melt, which can explain the unusually low lunar monthly tidal Q of ~30.