The uniqueness of the Fisher metric as information metric

Lê, Hông Vân

doi:10.1007/s10463-016-0562-0

Cited by 28 publications

(13 citation statements)

References 10 publications

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“…6. In [7,Chapter 3] and in [29,Definition 4.10] we propose different refinements of the notion of a k-integrable parametrized measure model, for which the validity of the condition (2) for k ≥ 1 implies the validity of the condition (2) for all 1 ≤ p ≤ k. Thought the present notion of a k-integrable parametrized measure model is not as elegant as we wish, it seems to us closest to suggestions of Amari and Cramer, see Remark 2.7.…”

Section: Parametrized Measure Modelsmentioning

confidence: 99%

Information geometry and sufficient statistics

Jost

Lê

et al. 2014

Probab. Theory Relat. Fields

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113

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Abstract. 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 notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the AmariChentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows. MSC2010: 53C99, 62B05

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Section: Parametrized Measure Modelsmentioning

confidence: 99%

Information geometry and sufficient statistics

Jost

Lê

et al. 2014

Probab. Theory Relat. Fields

Self Cite

113

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“…This notion was introduced by Chentsov in the case of finite sample spaces [12], but the natural generalization in Definition 2.2 to arbitrary sample spaces has been treated in [4], [5] and [17].…”

Section: Preliminary Resultsmentioning

confidence: 99%

Congruent Families and Invariant Tensors

Schwachhöfer

Jost

et al. 2018

Information Geometry and Its Applications

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Classical results of Chentsov and Campbell state that -up to constant multiples -the only 2-tensor field of a statistical model which is invariant under congruent Markov morphisms is the Fisher metric and the only invariant 3-tensor field is the Amari-Chentsov tensor. We generalize this result for arbitrary degree n, showing that any family of n-tensors which is invariant under congruent Markov morphisms is algebraically generated by the canonical tensor fields defined in [5].

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“…In general, since coordinate transformations preserve the probabilities associated to a joint proposition space, they also preserve several structures derived from them. One of these is the Fisher-Rao (information) metric [25,31,32], which was proved byČencov [26] to be the unique metric on statistical manifolds that represents the fact that the points ρ ∈ ∆ are probability distributions and not structureless [10] (For a summary of various derivations of the information metric, see [10] Section 7.4).…”

Section: Type I: Coordinate Transformationsmentioning

confidence: 99%

“…which could potentially depend on the entire distribution ρ as well as the point x d ∈ X . We impose DC1 by constraining (32) to only depend on the probabilities within the subdomain D since the variation (32) should not cause changes to the amount of correlations in the complementD, i.e.,…”

Section: Locality-dc1mentioning

confidence: 99%

The Design of Global Correlation Quantifiers and Continuous Notions of Statistical Sufficiency

Carrara

Vanslette

2020

Entropy

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Using first principles from inference, we design a set of functionals for the purposes of ranking joint probability distributions with respect to their correlations. Starting with a general functional, we impose its desired behavior through the Principle of Constant Correlations (PCC), which constrains the correlation functional to behave in a consistent way under statistically independent inferential transformations. The PCC guides us in choosing the appropriate design criteria for constructing the desired functionals. Since the derivations depend on a choice of partitioning the variable space into n disjoint subspaces, the general functional we design is the n-partite information (NPI), of which the total correlation and mutual information are special cases. Thus, these functionals are found to be uniquely capable of determining whether a certain class of inferential transformations, ρ → ∗ ρ ′ , preserve, destroy or create correlations. This provides conceptual clarity by ruling out other possible global correlation quantifiers. Finally, the derivation and results allow us to quantify non-binary notions of statistical sufficiency. Our results express what percentage of the correlations are preserved under a given inferential transformation or variable mapping.

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The uniqueness of the Fisher metric as information metric

Cited by 28 publications

References 10 publications

Information geometry and sufficient statistics

Information geometry and sufficient statistics

Congruent Families and Invariant Tensors

The Design of Global Correlation Quantifiers and Continuous Notions of Statistical Sufficiency

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