We applied Bayesian techniques to the problem of inferring the magnetic field strength in transversely oscillating solar coronal loops from observed periods and damping times. This was done by computing the marginal posterior probability density 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 observed waves can be modelled as standing or propagating magnetohydrodynamic (MHD) kink modes of magnetic flux tubes. Our results indicate that the magnetic field strength can be inferred, even if the densities inside and outside the structure are largely unknown. When information on plasma density is available, the method enables to self-consistently include this knowledge to further constrain the inferred magnetic field strength. The inclusion of the observed oscillation damping enables to obtain information on the transverse density structuring and considerably alters the obtained posterior for the magnetic field strength.