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|eprint=2312.09852
|year=2023
|class=cs.LG
}}</ref> where the more general case of non-isometrically embedded [[Riemannian manifold|Riemann manifolds]] is also treated. Here we restrict attention to [[isometry|isometrically]] embedded manifolds.
As running examples of manifolds with smooth, isometric embedding in <math>\R^n</math> we shall use:
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|first1=Niko
|last2=van Leeuwen
|first2=D.
|title=On calibration of language recognition scores
|book-title=Proceedings of IEEE Odyssey: The Speaker and Language Recognition Workshop
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|eprint=2408.02841
|year=2024
|class=stat.ML
}}</ref> uses the [[softmax function]] to renormalize categorical distributions after scaling and translation of the input distributions in log-probability space. For <math>\mathbf p, \mathbf q\in\Delta^{n-1}</math> and with parameters, <math>a\ne0</math> and <math>\mathbf c\in\R^n</math> the transform can be specified as:
:<math>
\mathbf q=f_\text{cal}(\mathbf p; a, \mathbf c) = \operatorname{softmax}(a^{-1}\log\mathbf p+\mathbf c)\;\iff\;
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In fact, the set of <math>f_\text{gcal}</math> transformations form a [[group mathematics|group]] under function composition. The set of <math>f_\text{cal}</math> transformations form a subgroup.
Also see: '''Dirichlet calibration''',<ref>{{cite
| title = Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration
| author = Meelis Kull, Miquel Perelló‑Nieto, Markus Kängsepp, Telmo Silva Filho, Hao Song, Peter A. Flach
|
| date = 28 October 2019
| class = cs.LG
}}</ref> which generalizes <math>f_\text{gcal}</math>, by not placing any restriction on the matrix, <math>\mathbf A</math>, so that invertibility is not guaranteed. While Dirichlet calibration is trained as a discriminative model, <math>f_\text{gcal}</math> can also be trained as part of a generative calibration model.
===Differential volume ratio for curved manifolds===
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|pages=573–586
|year=1994
|doi=10.1007/BF00773518
}}
</ref>
:<math>
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</math>
This result can be derived indirectly via the '''Angular central Gaussian distribution (ACG)''',<ref>
{{cite journal|title=Statistical analysis for the angular central Gaussian distribution on the sphere|last1=Tyler|first1=David E|journal=Biometrika|volume=74|number=3|pages=579–589|year=1987|doi=10.2307/2336697|jstor=2336697 }}
</ref> which can be obtained via normalized linear transform of either Gaussian, or uniform spherical variates. The first relationship can be used to derive the ACG density by a marginalization integral over the radius; after which the second relationship can be used to factor out the differential volume ratio. For details, see [[ACG distribution]].
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