In a Search for a Structure, Part 1: On Entropy

Gromov, Mikhael

doi:10.4171/120-1/4

Cited by 30 publications

(51 citation statements)

References 18 publications

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“…Those criticisms are used for highlight the lack of both Structure and Relations. Those criticisms also highlight the search of M. Gromov [15]. The need for structures and relations was the intuition of Peter McCullagh.…”

Section: What Is a Statistical Modelmentioning

confidence: 99%

“…Currently, the interest in information geometry is increasing. This comes from the links with many major research domains [14][15][16]. We address some significant aspects of those links.…”

Section: Some Explicit Formulasmentioning

confidence: 99%

“…[30] A few years after Misha Gromov raised a similar request: The Search of Structure. Fisher Information [15,16]. Those two titles are two formulations of the same need.…”

Section: Ige Stands For Information Geometrymentioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, "dualistic relation of Amari" and "Riemannian or symplectic Foliations"; "Koszul geometry" and "linearization of webs"; "KV homology" and "complexity of models". Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.

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Section: What Is a Statistical Modelmentioning

confidence: 99%

“…Currently, the interest in information geometry is increasing. This comes from the links with many major research domains [14][15][16]. We address some significant aspects of those links.…”

Section: Some Explicit Formulasmentioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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“…Since the group Gro(P) is isomorphic to the multiplicative group of positive real numbers [13]-this is a reformulation of the Bernoulli law of large numbers -the Grothendieck class [P ] Gro can be identified with exp ent(P ).…”

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confidence: 99%

“…For instance, just to warm up, one may start by elaborating on the category theoretic definition of the entropy suggested "In a Search for a Structure, Part 1: On Entropy" [13], where the entropy of a finite probability space P = {p i }, p i > 0, ∑ i p i = 1, comes as the class [P ] Gro of P in the Grothendieck group Gro(P) of the topological category P of finite probability spaces P and probability/measure preserving maps P → Q with a properly defined topological structure in P.…”

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confidence: 99%

Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014

Gromov

2015

Entropy

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Abstract:In this discussion, we indicate possibilities for (homological and non-homological) linearization of basic notions of the probability theory and also for replacing the real numbers as values of probabilities by objects of suitable combinatorial categories.Keywords: entropy; Bernoulli approximation; homology measuresThe success of the probability theory decisively, albeit often invisibly, depends on symmetries of systems this theory applies to. For instance:• The symmetry group of a single round of gambling with three dice has order 288 = 6 × 6 × 8: it is a semidirect product of the permutation group S 3 of order 6 and the symmetry group of the 3d cube, that is, in turn, is a semidirect product of S 3 and {±1} 3 .• The Bernoulli spaces (gp, p 1−p ) Z , 0 < p < 1, of (g, p)-sequences indexed by integers z ∈ Z = {⋯, −2, −1, 0, 1, 2, ⋯} are acted upon by a semidirect product of the infinite permutation group S ∞=Z ⊃ Z = {⋯, −2, −1, 0, 1, 2, ⋯} and the (compact) group {±1} Z = { g ↔ p} Z , with the role of the latter being essential even for p ≠ 1 2 where the probability measure is not preserved.• The system of identical point-particles • i in the Euclidean 3-space R 3 , that are indexed by a countable set I ∋ i, is acted upon by the isometry group of R 3 times the infinite permutation group S ∞=I .

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A Categorical Approach to Statistical Mechanics

Sergeant-Perthuis

2023

Lecture Notes in Computer Science

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In this document, accepted for the Twelfth Symposium on Compositional Structures (SYCO 12) [1], we aim to gather various results related to a compositional/categorical approach to rigorous Statistical Mechanics [2][3][4][5][6][7][8][9]. Rigorous Statistical Mechanics is centered on the mathematical study of statistical systems. Central concepts in this field have a natural expression in terms of diagrams in a category that couples measurable maps and Markov kernels [8]. We showed that statistical systems are particular representations of partially ordered sets (posets), that we call A -specifications, and expressed their phases, i.e., Gibbs measures, as invariants of these representations. It opens the way to the use of homological algebra to compute phases of statistical systems. Two central results of rigorous Statistical Mechanics are, firstly, the characterization of extreme Gibbs measure as it relates to the zero-one law for extreme Gibbs measures, and, secondly, their variational principle which states that for translation invariant Hamiltonians, Gibbs measures are the minima of the Gibbs free energy. We showed in [9] how the characterization of extreme Gibbs measures extends to A -specifications; we proposed in [10] an Entropy functional for A -specifications and gave a message-passing algorithm, that generalized the belief propagation algorithm of graphical models (see O. Peltre's SYCO 8 talk, [11] and [12]), to minimize variational free energy.

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In a Search for a Structure, Part 1: On Entropy

Cited by 30 publications

References 18 publications

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Symmetry, Probabiliy, Entropy: Synopsis of the Lecture at MAXENT 2014

A Categorical Approach to Statistical Mechanics

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