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

05/07/2023

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by   Kaori Yamaguchi, et al.

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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-Lê-Schwachhöfer and Fisher-Neyman for sufficient statistics.