Invariant Geometric Structures on Statistical Models
We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov , Amari  and Pistone-Sempi . We give a complete description of n-tensor fields that are invariant under sufficient statistics. In the cases \(n= 2\) and \(n = 3\), the only such tensors are the Fisher metric and the Amari-Chentsov tensor. While this has been shown by Chentsov  and Campbell  in the case of finite measure spaces, our approach allows to generalize these results to the cases of infinite sample spaces and arbitrary n. Furthermore, we give a generalisation of the monotonicity theorem and discuss its consequences.
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