  The quest for precision in parameter estimation is a fundamental task in
different scientific areas. The relevance of this problem thus provided the
motivation to develop methods for the application of quantum resources to
estimation protocols. Within this context, Bayesian estimation offers a
complete framework for optimal quantum metrology techniques, such as adaptive
protocols. However, the use of the Bayesian approach requires extensive
computational resources, especially in the multiparameter estimations that
represent the typical operational scenario for quantum sensors. Hence, the
requirement to characterize protocols implementing Bayesian estimations can
become a significant challenge. This work focuses on the crucial task of
robustly benchmarking the performances of these protocols in both single and
multiple-parameter scenarios. By comparing different figures of merits,
evidence is provided in favor of using the median of the quadratic error in the
estimations in order to mitigate spurious effects due to the numerical
discretization of the parameter space, the presence of limited data, and
numerical instabilities. These results, providing a robust and reliable
characterization of Bayesian protocols, find natural applications to practical
problems within the quantum estimation framework.
