Liquid water was present on the surface of Mars in the distant past; part of that water is now in the ground in the form of permafrost and heat from the molten interior of the planet could cause it to melt at depth. MARSIS (Mars Advanced Radar for Subsurface and Ionosphere Sounding) has surveyed the Martian subsurface for more than fifteen years in search for evidence of such water buried at depth. Radar detection of liquid water can be stated as an inverse electromagnetic scattering problem, starting from the echo intensity collected by the antenna. In principle, the electromagnetic problem can be modelled as a normal plane wave that propagates through a three-layered medium made of air, ice and basal material, with the final goal of determining the dielectric permittivity of the basal material. In practice, however, two fundamental aspects make the inversion procedure of this apparent simple model rather challenging: i) the impossibility to use the absolute value of the echo intensity in the inversion procedure; ii) the impossibility to use a deterministic approach to retrieve the basal permittivity. In this paper, these issues are faced by assuming a priori information on the ice electromagnetic properties and adopting an inversion probabilistic approach. All the aspects that can affect the estimation of the basal permittivity below the Martian South polar cap are discussed and how detection of the presence of basal liquid water was done is described.