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Cruising Cruising the Hypothesis Space: the Hypothesis Space: Information Information Geometry Geometry ( with pictures ) ( with pictures )

By Carlos C. Rodriguez

Dept. of MathematicsThe University at Albanycarlos@math.albany.edu

•One

•Infinity

Geometry?Geometry?

Cross-Ratio of 4 PointsCross-Ratio of 4 Points

Klein’s Erlanger Klein’s Erlanger ProgrammeProgramme

The Theorem Between The Theorem Between Theories and ThingsTheories and Things

• Data, D (facts or things)• Hypothesis, H (theories)• Prior info, O (the rest)

Likelihood Ratio is a Likelihood Ratio is a Cross-Ratio!Cross-Ratio!

Omega.albany.edu:8008/

Information Geometry ? Information Geometry ?

Models as PlacesModels as Places

• Parameters are Parameters are coordinatescoordinates

• Tangent vectors Tangent vectors are signed are signed measuresmeasures

• Tangent space at Tangent space at is a subspace of is a subspace of L²(PL²(P))

P

P(t)

lim (P(t) - P)/t = Q t 0

has density w.r.t. P

dQdP

= v ln f (x| )

j

j

The Tangent SpaceThe Tangent Space

inner product of

tangent vector

Fisher information

implies

Entropy and GeometryEntropy and Geometry

tv

v

family of lengths family of lengths

Riemannian lengthRiemannian length

Family of geodesicsFamily of geodesics

-deviation

in direction v

-deviation

in direction v

Entropy and IgnoranceEntropy and Ignorance• Let

• Then

• produces the family of entropic priors• produces the family of entropic priors

• volume element Jeffreys priorJeffreys prior

Entropy and TimeEntropy and Time

4

Entropy and TimeEntropy and Time

Radially Symmetric Radially Symmetric DistributionsDistributions

• location parameter • scale parameter • shape parameter

• dimensional manifold

• For fix

• curvature -1/J()

Uncertain Spinning SpaceUncertain Spinning Space Let be radially symmetric about 0 and

let i be the pseudo-scalar of 3-space. Then

y = + i Represents a location + an uncertain degree of orientation

or spinning with direction and magnitude given by i is a constant of magnitude 1.

i commutes with everything. i is an oriented unit volume. i ²= -1

• E(y) = • E(y²) = ² - ²

Spacetime as the Spacetime as the Hypothesis Space of Hypothesis Space of Radially Symmetric Radially Symmetric DistributionsDistributions

• Space is dim and only on the average appears as 4 dim spacetime

•Time is a consequence of uncertain space.

• Spin is a property of space so all fundamental particles must have spin

Curvature and InformationCurvature and Information

Mass-Energy curves spacetime

Prior information curves hypothesis spaces Prior information curves hypothesis spaces

SUNY at Albany

In Closing...In Closing...

• Hypothesis spaces are cruisable.Hypothesis spaces are cruisable.

• Evidence is accumulating that Evidence is accumulating that statistical inference may include GR. statistical inference may include GR.

And, there is a lot to be done:

• The role of curvature in statistics.The role of curvature in statistics.

• A new approach to Quantum Gravity.A new approach to Quantum Gravity.

• Hypothesis spaces are cruisable.Hypothesis spaces are cruisable.

• Evidence is accumulating that Evidence is accumulating that statistical inference may include GR. statistical inference may include GR.

And, there is a lot to be done:

• The role of curvature in statistics.The role of curvature in statistics.

• A new approach to Quantum Gravity.A new approach to Quantum Gravity.