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

Abstract

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 .

Topological Information Data Analysis: Poincare-Shannon Machine and Statistical Physic of Finite Heterogeneous Systems

“…We underline that no stationarity, ergodicity or Markovian (and all the more iid memoryless processes) assumptions have been made in the formalism. Altogether, the formalization of information paths provides an elementary setting to tackle the problematics raised by Gromov on symmetry, probability, entropy [124]. Question: recently Baez and Fong published a Noether Theorem for Markov Processes [125]; can we derive a Noether theorem for random discrete processes in general, that is for all the symmetric group S n using the present construction?…”

The Gauge Equation in Statistical Manifolds: An Approach through Spectral Sequences

“…We recall that a statistical manifold is a quadruple (M, g, ∇, ∇) such that ∇ and ∇ + are dual connections with respect to the metric g which are both torsion-free. These structures are of the utmost importance in information geometry [10,11], and are named after their appearance in statistical problems [12].…”

The τq-Fourier transform: Covariance and uniqueness

“…An important difference of all these treatments with the current one, is that as in the case of modern Algebraic Geometry (see, for instance, [18]) following pioneering ideas and proposals of A. Grothendieck, one has to allow for changes of the underlying field in all constructions, thus needing definitions which are "covariant" in a categorical sense. Indeed, there is suspicion that categorical constructions can be at the core of the formulation, and may play a considerable role in establishing and illuminating properties of the BGS entropy pertinent to physical systems [19,20].…”

“…In (20), τ * q is the induced map from τ q to the topological dual space of R with respect to the Fourier pairing, the dual being also R. Notice that diagram ( 20) is heuristic; its role is to convey the idea of the τ q -Fourier transform versus the usual Fourier transform F, rather than being precise. In a more careful treatment, as in Subsection 3.2 below, let S q and S ′ q indicate the spaces of Schwartz functions and of tempered distributions over R q respectively.…”