  In topological data analysis (TDA), persistence diagrams have been a
succesful tool. To compare them, Wasserstein and Bottleneck distances are
commonly used. We address the shortcomings of these metrics and show a way to
investigate them in a systematic way by introducing bottleneck profiles. This
leads to a notion of discrete Prokhorov metrics for persistence diagrams as a
generalization of the Bottleneck distance. They satisfy a stability result and
bounds with respect to Wasserstein metrics. We provide algorithms to compute
the newly introduced quantities and end with an discussion about experiments.
