Tensor train (TT) decomposition, a powerful tool for analyzing multidimensional data, exhibits superior performance in many signal processing and machine learning tasks. However, existing TT decomposition methods either require the knowledge of the true TT ranks, or extensive fine-tuning of the balance between model complexity and representation accuracy. In this paper, a fully Bayesian treatment of TT decomposition is employed to enable automatic rank determination. Based on the proposed probabilistic model, an efficient learning algorithm is derived under the variational inference framework. Simulation results on synthetic data show the success of the proposed model and algorithm in recovering the ground-truth TT structure from noisy data.