It is the main purpose of this study to examine the deeper structure of the Moon in the light of four numbers. These are lunar mass M, mean moment of inertia I, second degree tidal Love number k2, and the quality factor Q, accounting for tidal dissipation within the solid body of the Moon. The former two have been measured by Lunar Prospector to high precision, and more than 30 years of lunar laser ranging (LLR) data have led to an estimate of the second degree tidal Love number and quality factor. The inverse problem dealt with here of obtaining information on the lunar density and S wave velocity profile from the four numbers follows our earlier investigations by employing an inverse Monte Carlo sampling method. We present a novel way of analyzing the outcome using the Bayes factor. The advantage lies in the fact that rather than just looking at a subset of sampled models, we investigate all the information sampled in different runs, i.e., take into account all samples, in order to estimate their relative plausibility. The most likely outcome of our study, based on the data, their uncertainties, and prior information, is a central core with a most probable S wave velocity close to 0 km/s, density of ~7.2 g/cm3 and radius of about 350 km. This is interpreted as implying the presence of a molten or partially molten Fe core, in line with evidence presented earlier using LLR regarding the dissipation within the Moon.