Estimation of quenched random fields in the inverse Ising problem using a diagonal matching method

We consider a method for the accurate estimation of quenched random fields in the inverse Ising problem. Approximations such as the mean-field or Bethe methods are applied to estimate quenched random coupling parameters and external fields. A diagonal matching method is introduced to ensure consistency of the diagonal part of the susceptibility, and the method yields an accurate estimation of the external fields. We introduce the diagonal matching method into the mean-field, Thouless-Anderson-Palmer, and Bethe approximations, and we investigate the effect of the diagonal matching method on the accuracy of estimation of the external fields.