Abstract
We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and Pistone-Sempi [10]. 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 [7] and Campbell [5] 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.
L. Schwachhöfer—speaker
J. Jost—partially supported by ERC Advanced Grant FP7-267087
H.V. Lê—partially supported by RVO: 6798584.
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Schwachhöfer, L., Ay, N., Jost, J., Vân Lê, H. (2015). Invariant Geometric Structures on Statistical Models. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_17
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DOI: https://doi.org/10.1007/978-3-319-25040-3_17
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