Let $(M,g)$ be an infinite dimensional statistical manifold with the Fisher information metric $g$. Is it true that any isometry on this manifold must correspond to a sufficient statistic?
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$\begingroup$ I don't know an answer, but I wonder if you have a few concrete examples? $\endgroup$– Michael HardyCommented Jan 17, 2014 at 3:22
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$\begingroup$ I'm thinking of the manifold $N(\mu,\sigma^2)$. It's the upper half-plane, with metric something like $\displaystyle\sqrt{d\mu^2+\frac{d\sigma^2}{\sigma^2}}$? The minimal sufficient statistic for an i.i.d. sample $X_1,\ldots,X_n$ is just the pair $(X_1+\cdots+X_n,X_1^2+\cdots+X_n^2)$. How would this correspond to some isometry? $\endgroup$– Michael HardyCommented Jan 17, 2014 at 3:27
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