The possibility of inferring the properties of electron density fluctuations in tokamak plasmas from a reflectometer signal by means of Bayesian methods is investigated. Within the physical optics approximation, the interaction of the probing beam with the plasma is described as reflection from a surface with stochastic properties that is simulated numerically. A Bayesian technique is outlined to solve the inverse problem to determine the surface characteristics from the power spectrum of the reflectometer signal. It is shown that satisfactory estimates of the length and timescales and the amplitude of density fluctuations can be obtained in conditions relevant to core tokamak plasmas.