The aim of operational risk modeling is to provide a reasonably accurate, reasonably precise and reasonably robust estimation of capital requirements, including a level of sensitivity that is consistent with the changes of the risk profile. A way to obtain robust capital estimates is through optimal B-robust (OBR) methods. Previous research has shown that OBR methods might mitigate the bias in capital risk quantification when compared with classical maximum likelihood estimation. Motivated by requirements related to operational risk measurement, the aim of this work is to integrate prior information into a robust parameter estimation framework via OBRestimating functions. Unfortunately, the evaluation of OBR-estimating functions for different parameter values is cumbersome, and this rules out the use of many pseudo-likelihood methods. To deal with this issue, we suggest resorting to approximate Bayesian computation (ABC) machinery, using the OBR-estimating function as the summary statistic. Unlike other methods, the proposedABC-OBR algorithm requires the evaluation of the OBR-estimating function at a fixed parameter value but using different data samples, which is computationally trivial. The method is illustrated using a small simulation study and applications to two real operational risk data sets.