Context. Time and spatial damping of transverse magnetohydrodynamic (MHD) kink oscillations is a source of information on the cross-field variation of the plasma density in coronal waveguides. Aims: We show that a probabilistic approach to the problem of determining the density structuring from the observed damping of transverse oscillations enables us to obtain information on the two parameters that characterise the cross-field density profile. Methods: The inference is performed by computing the marginal posterior distributions for density contrast and transverse inhomogeneity length-scale using Bayesian analysis and damping ratios for transverse oscillations under the assumption that damping is produced by resonant absorption. Results: The obtained distributions show that, for damping times of a few oscillatory periods, low density-contrasts and short inhomogeneity length scales are more plausible to explain observations. Conclusions: This means that valuable information on the cross-field density profile can be obtained even if the inversion problem, with two unknowns and one observable, is a mathematically ill-posed problem.