Inversion Analyses Based on ABIC with Non-full Rank Prior Information

In inversion analyses using ABIC, a non-full rank matrix for prior constraints is allowed in the formulation of Yabuki and Matsu'ura (1992), while Fukuda and Johnson (2008) claimed that the matrix must be full rank. This problem depends on consideration about the value of “zero” of zero eigenvalues contained in the prior constraint matrix. In actual inversion analyses, we must have some prior information about model parameters, even if it is not explicitly expressed. Therefore, the “zero” of the zero eigenvalues is considered to be not zero exactly. Based on this consideration, we can obtain the same inversion solution as Yabuki and Matsu'ura (1992). Mathematical difficulty in expressing the prior probability density function for a matrix with zero eigenvalues can be avoided by the use of non-informative prior.