A variational Bayesian approach to robust identification of switched auto-regressive exogenous models is developed in this paper. By formulating the problem of interest under a full Bayesian identification framework, the number of local-models can be determined automatically, while accounting for the uncertainty of parameter estimates in the overall identification procedure. A set of significance coefficients is used to assign proper importance weights to local-models. By maximizing the marginal likelihood of the identification data, insignificant local-models will be suppressed and the optimal number of local-models can be determined. Considering the fact that the identification data may be contaminated with outliers, t distributions with adjustable tails are utilized to model the contaminating noise so that the proposed identification algorithm is robust. The effectiveness of the proposed Bayesian approach is demonstrated through a simulated example as well as a detailed industrial application.