I will explain how to use persistent homology in a supervised way, allowing to optimize over various models for the observed homological information. The focus is on studying the space of stable translations from homological information into information that can be analysed through more basic operations such as counting and integration enabling the use of statistical tools to its outcomes. I will explain how a pseudometric on the set of tame parametrized vector spaces, which is a natural place where homological invariants of data live, leads to a stable translation. Every such pseudometric gives therefore a model for extracting information from persistent homology. Fitting this model to the training data is what in our approach persistence based supervised learning is.