# Chentsov’s theorem for exponential families

@article{Dowty2017ChentsovsTF, title={Chentsov’s theorem for exponential families}, author={James G. Dowty}, journal={Information Geometry}, year={2017}, volume={1}, pages={117-135} }

Chentsov’s theorem characterizes the Fisher information metric on statistical models as the only Riemannian metric (up to rescaling) that is invariant under sufficient statistics. This implies that each statistical model is equipped with a natural geometry, so Chentsov’s theorem explains why many statistical properties can be described in geometric terms. However, despite being one of the foundational theorems of statistics, Chentsov’s theorem has only been proved previously in very restricted… Expand

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#### References

SHOWING 1-10 OF 19 REFERENCES

Information geometry and sufficient statistics

- Mathematics
- 2012

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari–Chentsov tensor. In statistics, the… Expand

An extended Čencov characterization of the information metric

- Mathematics
- 1986

Cencov has shown that Riemannian metrics which are derived from the Fisher information matrix are the only metrics which preserve inner products under certain probabilistically important mappings. In… Expand

An Infinite-Dimensional Geometric Structure on the Space of all the Probability Measures Equivalent to a Given One

- Mathematics
- 1995

Let M μ be the set of all probability densities equivalent to a given reference probability measure μ. This set is thought of as the maximal regular (i.e., with strictly positive densities)… Expand

Geometrical Foundations of Asymptotic Inference

- Mathematics
- 1997

Overview and Preliminaries. ONE-PARAMETER CURVED EXPONENTIAL FAMILIES. First-Order Asymptotics. Second-Order Asymptotics. MULTIPARAMETER CURVED EXPONENTIAL FAMILIES. Extensions of Results from the… Expand

The uniqueness of the Fisher metric as information metric

- Mathematics
- 2013

We define a mixed topology on the fiber space $$\cup _\mu \oplus ^n L^n(\mu )$$∪μ⊕nLn(μ) over the space $${\mathcal M}({\Omega })$$M(Ω) of all finite non-negative measures $$\mu $$μ on a separable… Expand

Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice

- Mathematics
- 2001

Preface. Acknowledgments. INTRODUCTION. Random Vectors. Linear Operators. Infinitely Divisible Distributions and Triangular Arrays. MULTIVARIATE REGULAR VARIATION. Regular Variations for Linear… Expand

Uniqueness of the Fisher-Rao metric on the space of smooth densities

- Mathematics
- 2014

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism… Expand

Introduction to the theory of distributions

- Mathematics
- 1982

1. Test functions and distributions 2. Differentiation and multiplication 3. Distributions and compact support 4. Tensor products 5. Convolution 6. Distribution kernels 7. Co-ordinate transforms and… Expand

Fisher information and stochastic complexity

- Mathematics, Computer Science
- IEEE Trans. Inf. Theory
- 1996

A sharper code length is obtained as the stochastic complexity and the associated universal process are derived for a class of parametric processes by taking into account the Fisher information and removing an inherent redundancy in earlier two-part codes. Expand

The Minimum Description Length Principle in Coding and Modeling

- Computer Science, Mathematics
- IEEE Trans. Inf. Theory
- 1998

The normalized maximized likelihood, mixture, and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms. Expand