Omega.albany.edu:8008/ Cruising the Hypothesis Space: Information Geometry ( with pictures ) By Carlos C. Rodriguez Dept. of Mathematics The University at Albany

One Infinity Geometry?

Cross-Ratio of 4 Points

Klein’s Erlanger Programme

The Theorem Between Theories and Things Data, D (facts or things) Hypothesis, H (theories) Prior info, O (the rest)

Likelihood Ratio is a Cross-Ratio!

Omega.albany.edu:8008/ Information Geometry ?

Models as Places Parameters are coordinatesParameters are coordinates Tangent vectors are signed measuresTangent vectors are signed measures Tangent space at  is a subspace of L²(P  )Tangent space at  is a subspace of L²(P  ) PP P  (t) lim (P  (t) - P  )/t = Q t  0  has density w.r.t. P  dQ dP   =   v    ln f ( x|  )    j j

The Tangent Space inner product of tangent vector Fisher information implies

Entropy and Geometry  tv   v  family of lengths Riemannian length  Family of geodesics  -deviation in direction v  -deviation in direction v

Entropy and Ignorance Let Then produces the family of entropic priors volume element Jeffreys prior

Entropy and Time

Radially Symmetric Distributions location parameter  scale parameter  shape parameter   dimensional manifold For fix  curvature -1/J(  )

Uncertain 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 Hypothesis Space of Radially Symmetric Distributions 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 Information Mass-Energy curves spacetime Prior information curves hypothesis spaces

SUNY at Albany In Closing... Hypothesis spaces are cruisable.Hypothesis spaces are cruisable. Evidence is accumulating that statistical inference may include GR.Evidence is accumulating that 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 statistical inference may include GR.Evidence is accumulating that 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.