For inversion analysis using Akaike's Bayesian Information Criterion (ABIC) we obtained a proper formulation to incorporate two sorts of partially-dependent prior information into observed data by Bayes' rule. In previous improper formulations, complete independence of the two sorts of prior information has been assumed. Through comparison of the solutions obtained by the improper formulation with those by the proper formulation, we demonstrated the validity of the proper formulation. The results show that the previous improper formulation gives proper solutions, if and only if the data used for inversion are accurate and sufficient. If the data are inaccurate and/or insufficient, the improper formulation gives somewhat biased and oversimplified solutions. From a poor data set, if we employed the improper formulation, we could not find any solution because of the vanishing of the local minimum of ABIC.