Variational nonlinear chirp mode decomposition (VNCMD) is a recently introduced method for nonlinear chirp signal decomposition that has aroused notable attention in various fields. One limiting aspect of the method is that its performance relies heavily on the setting of the bandwidth parameter. To overcome this problem, we here propose a Bayesian implementation of the VNCMD, which can adaptively estimate the instantaneous amplitudes and frequencies of the nonlinear chirp signals, and then learn the active dictionary in a data-driven manner, thereby enabling a high-resolution time-frequency representation. Numerical example of both simulated and measured data illustrate the resulting improvement performance of the proposed method.