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I'm wondering how to go from: $$ \mathrm{Exponential~ families} \implies \mathrm{Kähler~manifolds} $$

I read that the tangent bundle of an exponential family naturally forms a Kähler manifold. I also found that for any exponential family, one can find a corresponding Kähler manifold from it. This means one can study "statistical mirror symmetry" prevalent in algebraic geometry and theoretical physics, where Calabi-Yau manifolds play a big part.

See this youtube video for some more discussion and context.

In information geometry, there are several examples of exponential families that naturally exhibit hyperbolic geometry, notably the Gaussian distribution. But I don't see how to go further and connect exponential families to Kähler manifolds.

For a given distribution that is part of the exponential family, how does its tangent bundle naturally form a Kähler manifold?

I'm fine with a high-level description of this so that I can get some intuition.

J. Zimmerman
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  • @MarianoSuárez-Álvarez The audience should be able to sketch how to go from a distribution that is in the exponential family and obtain a Kähler manifold. I suppose it's mainly for an information geometry audience – J. Zimmerman Aug 04 '23 at 21:04
  • it would help if you added references: where did you read it? in what context? – Johnny Lemmon Sep 10 '23 at 13:04
  • Sure I will update in the next few hours – J. Zimmerman Sep 10 '23 at 15:11
  • You should not expect readers here to watch a Youtube video in order to understand your question. Furthermore, lack of responses to your question should make it clear that the expected audience for your question simply does not exist on MSE. – Moishe Kohan Sep 27 '23 at 15:00
  • I know nothing about the subject (exponential families), but googling yields this reference connecting exponential families and Kahler geometry: https://arxiv.org/abs/1203.2056. – Moishe Kohan Sep 27 '23 at 15:06

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