We present a method to assess the plausibility of alternative mechanisms to explain the damping of magnetohydrodynamic transverse waves in solar coronal loops. The considered mechanisms are resonant absorption of kink waves in the Alfvén continuum, phase mixing of Alfvén waves, and wave leakage. Our methods make use of Bayesian inference and model comparison techniques. We first infer the values for the physical parameters that control the wave damping, under the assumption of a particular mechanism, for typically observed damping timescales. Then, the computation of marginal likelihoods and Bayes factors enable us to quantify the relative plausibility between the alternative mechanisms. We find that, in general, the evidence is not large enough to support a single particular damping mechanism as the most plausible one. Resonant absorption and wave leakage offer the most probable explanations in strong damping regimes, while phase mixing is the best candidate for weak/moderate damping. When applied to a selection of 89 observed transverse loop oscillations, with their corresponding measurements of damping timescales and taking into account data uncertainties, we find that positive evidence for a given damping mechanism is only available in a few cases.