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I consider a statistical manifold equipped with the Fisher Information Metric.

I want to show that for the exponential family (with no additional constraint), the Hellinger distance coincides with the Fisher-Rao distance, which is the geodesic distance coming from the Fisher Information Metric (this is said to be true here : https://twitter.com/gabrielpeyre/status/1338000702627590145?lang=fr).

I am looking for a reference in which I can find the proof of this fact.

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