Amplitude variation with offset (AVO) inversion, particularly for more than two model parameters, is a highly ill-posed problem and, hence, regularization is indispensable. Here, we propose a regularized inverse problem to mitigate the ill-posedness of the amplitude inversion. The regularization is added to measure the difference in information between the a priori probability density function and the predicted probability density of the inverted parameters. Information theory provides a collection of contrast functions which quantify the divergence from one probability distribution to another, such as the relative entropy. The a priori density is approximated by a Gaussian mixture model, obtained from well logs and rock physics model. The mixture model is a density estimator, providing the statistical properties of the model parameters of interest. The likelihood of the data and the divergence are combined in an augmented Lagrangian scheme, the alternating direction method of multipliers (ADMM), to obtain a unique solution that best generate the recorded seismic data and satisfy the geological constraints conveyed by the a priori probability density function. The proposed inversion scheme is then applied to the anisotropy AVO inversion, for estimating the elastic and seismic anisotropy parameters of shale formations. Compared to the unconstrained minimization, the P- and S-wave velocity, and $\varepsilon$ are better recovered, moreover, density and Thomsen’s $\delta$ are well-constrained.