Abstract
We provide an alternative differential geometric framework of the manifold \(\mathbb M\) of parametric statistical models. While adopting the Fisher-Rao metric as the Riemannian metric g on \(\mathbb M\), we treat the original parameterization of the statistical model as affine coordinate chart on the manifold endowed with a flat connection, instead of using a pair of torsion-free affine connections with generally non-vanishing curvature. We then construct its g-conjugate connection which, while necessarily curvature-free, carries torsion in general. So instead of associating a statistical structure to \(\mathbb M\), we construct a statistical manifold admitting torsion (SMAT). We show that \(\mathbb M\) is dually flat if and only if torsion of the conjugate connection vanishes.
The project is supported by DARPA/ARO Grant W911NF-16-1-0383 (“Information Geometry: Geometrization of Science of Information”, PI: Zhang).
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Zhang, J., Khan, G. (2019). New Geometry of Parametric Statistical Models. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_30
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DOI: https://doi.org/10.1007/978-3-030-26980-7_30
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