A classic application of coronal seismology uses transverse oscillations of waveguides to obtain estimates of the magnetic field strength. The procedure requires information on the density of the structures. Often it ignores the damping of the oscillations. We computed marginal posteriors for parameters such as the waveguide density, the density contrast, the transverse inhomogeneity length scale, and the magnetic field strength under the assumption that the oscillations can be modelled as standing magnetohydrodynamic (MHD) kink modes damped by resonant absorption. Our results show that the magnetic field strength can be properly inferred, even if the densities inside and outside the structure are largely unknown. Incorporating observational estimates of plasma density further constrains the obtained posteriors. The amount of information that is included a priori for the density and the density contrast influences their corresponding posteriors, but very little the inferred magnetic field strength. The decision to include or leave out the information on the damping and the damping timescales has a minimal impact on the obtained magnetic field strength. In contrast to the classic method, which provides numerical estimates with error bars or possible ranges of variation for the magnetic field strength, Bayesian methods offer the full distribution of plausibility over the considered range of possible values. The methods applied to available datasets of observed transverse loop oscillations can be extended to prominence fine structures or chromospheric spicules, and implemented to propagating waves in addition to standing oscillations.