Information geometry and classical Cramér–Rao-type inequalities

Mishra, Kumar Vijay; Kumar, M. Ashok

doi:10.1016/bs.host.2021.07.005

Cited by 2 publications

(2 citation statements)

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“…Following standard approaches in information geometry [1][2][3] the Bregman divergence D and dual Bregman divergence D * on N can be written as [1]…”

Section: Exponential Family Distributions and Bregman Geometrymentioning

confidence: 99%

“…Information geometry is the study of parameterized probability distributions using the machinery of differential and Riemannian geometry [1][2][3][4]. While information geometry has many statistical applications [5][6][7][8][9], it is also studied as a unique geometrical setting in its own right [10][11][12], and is intrinsically linked to statistical mechanics [13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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The Bayesian Central Limit Theorem for Exponential Family Distributions: A Geometric Approach

Goehle

2024

Preprint

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The Bernstein-von Mises theorem, also known as the Bayesian Central Limit Theorem (BCLT), states that under certain assumptions a posterior distribution can be approximated as a multivariate normal distribution as long as the number of pseudo-observations is large. We derive a form of the BCLT for the canonical conjugate prior of a regular exponential family distribution using the machinery of information geometry. Our approach applies the core approximation for the BCLT, Laplace's method, to the free-entropy (i.e., log-normalizer) of an exponential family distribution. Additionally, we formulate approximations for the Kullback-Leibler divergence and Fisher-Rao metric on the conjugate prior manifold in terms of corresponding quantities from the likelihood manifold. We also include an application to the categorical distribution and show that the free-entropy derived approximations are related to various series expansions of the gamma function and its derivatives. Furthermore, for the categorical distribution, the free-entropy approximation produces higher order expansions than the BCLT alone.

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“…Following standard approaches in information geometry [1][2][3] the Bregman divergence D and dual Bregman divergence D * on N can be written as [1]…”

Section: Exponential Family Distributions and Bregman Geometrymentioning

confidence: 99%