Bayesian model selection between two of the most commonly used circular models, namely, the von Mises distribution and the Wrapped Symmetric \(\alpha \) -Stable distribution is considered here. Our approach is based on posterior model probabilities and the corresponding posterior model odds, which are functions of Bayes factors. Marginal likelihoods under the two models are estimated based on prior distributions for the parameters that occur in these two competing models. The proposed methodology is analyzed and assessed through an extensive simulation study and shown to perform very well.