Page - 212 - in Differential Geometrical Theory of Statistics
Image of the Page - 212 -
Text of the Page - 212 -
Entropy2016,18, 433
subject to therequirement
D2 ∫
F a=0.
Definition57. AnMSE-fibration
[E,p]→ [M,D]
is calledanexponentialmodel if the followingconditionsare satisfied.
(1) ThebasemanifoldMsupports a locallyflat structure (M,∇)andarealvalued functionψ∈C∞(M).
(2) The total spaceE supports a realvaluedrandomfunction a.
(3) The triple [a,∇,ψ] is subject to the followingrequirement
(4) ∇2∫F(a)=O,
(5) p(e)= exp[a(e)−ψ(π(e))].
Remark5. AtonesideLocalizationsof exactHessianhomological statisticalmodelsyield the classicalKoszul
InformationGeometry [55]. That is but the classicalHessian InformationGeometry. Atanother side theKV
homologylearnsthat theHessian informationGeometry is thesamethinkas thegeometryof exponential famillies
seePartASection5,Theorem16.
Reminder.
In theAppendixAto thispaper the readerwillfindanewinvariant rb(p)measuringhowfar frombeing
anexponentialmodel is anMSE-fibration
[E,p]→ [M,D].
Bythevirtueof results inPartA, tobeanexponentialmodeldependsonhomological conditions.
8.4.1. TheEntropyFlow
Wearegoingto introduce thenotionof localentropyflow. Subsequentlywewill showthat the
Fisher informationofamodel [E,π,M,D,p] is theHessianof the localentropyflow.
TostartweconsideraMSE-fibration
[E,p]→ [M,D].
That isanotherpresentationof thestatisticalmodel [E,π,M,D,p].
Let [Uj,Φj×φj,γij,Pj]beanatlasof [E,π,M,D,p].Weput
[Θj,Pj]= [Φj(Ej),p◦Φ−1j ].
Thenevery [Θj,Pj] isa local statisticalmodel for (Ξ,Ω).
LeX,Ybe twovectorfieldsdefined inUj and letψX(t)andψY(s)be their localflowsdefined
inUj. Thenweset
Φj(e)= [θj(e),ξj(e)]= [φj(π(e)),ξj(e)], e∈Ej,
ψ˜j(t)[θj(e),ξj(e)]= ([φjψX(t)φ−1j ][θj(e)],ξj(e)),
ψ˜j(s)[θj(e),ξj(e)]= ([φjψY(s)φ−1j][θj(e)],ξj(e)).
Definition58. The local entropyflowof thepair (X,Y) is the functionEntjX,Y definedby
EntjX,Y(s,t)(π(e))= ∫
Ξ {
Pj[ψ˜X(s)(Φ(e))]log[Pj[ψ˜Y(t)(Φj(e))]] }
dξ(e).
212
Differential Geometrical Theory of Statistics
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik