On statistics which are almost sufficient from the viewpoint of the Fisher metrics

We introduce a quantitatively weak version of sufficient statistics such that the Fisher metric of the induced parametrized measure model is bi-Lipschitz equivalent to the Fisher metric of the original model. We characterize such statistics in terms of the conditional probability or by the existence of a certain decomposition of the density function in a way similar to characterizations of due to Ay-Jost-L\^e-Schwachh\"ofer and Fisher-Neyman for sufficient statistics.