**Stephen P. King** (*stephenk1@home.com*)

*Fri, 11 Jun 1999 19:55:05 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 404] On the Problem of Information Flow between LSs"**Previous message:**Stephen P. King: "[time 402] Re: [time 400] On the Problem of Information Flow between LSs"

Hi all,

I data-mined this:

http://www.mth.kcl.ac.uk/~streater/infolie.html

http://www.mth.kcl.ac.uk/~streater/inforel.html

http://www.mth.kcl.ac.uk/~streater/smoomani.html

http://www.mth.kcl.ac.uk/~streater/onsager.html

http://www.mth.kcl.ac.uk/~streater/soret.html

http://worldscientific.com/books/physics/p120.html

http://xxx.soton.ac.uk/abs/gr-qc/9701051

Title: Statistical Geometry

Authors: Dorje C. Brody and Lane P. Hughston

Comments: 27 pages, Rev Tex File, submitted to Proc. Roy. Soc. London

Report-no: Imperial/TP/95-96/42

\\

A statistical model M is specified by a family of probability

distributions,

characterised by a set of continuous parameters known as the parameter

space.

This possesses natural geometrical properties induced by the embedding

of the

family of probability distributions into the space of all

square-integrable

functions. More precisely, by consideration of the square-root density

function

we can regard M as a submanifold of the unit sphere S in a real Hilbert

space

H. Therefore, H effectively embodies the `state space' of the

probability

distributions, and the geometry of the given statistical model can be

described

in terms of the embedding of M in S. The geometry in question is

characterised

by a natural Riemannian metric (the Fisher-Rao metric), and as a

consequence

various aspects of classical statistical inference can be formulated in

a

natural geometric setting. In particular, we focus attention on the

variance

lower bounds for statistical estimation, and establish generalisations

of the

classical Cram\'er-Rao and Bhattacharyya bounds, described in terms of

the

geometry of the underlying real Hilbert space. The statistical model M

can then

be specialised to the case of a submanifold of the state space of a

quantum

mechanical system. This can be pursued by introducing a compatible

complex

structure on the underlying real Hilbert space, thus allowing the

operations of

ordinary quantum mechanics to be reinterpreted in the language of real

Hilbert

space geometry. The application of generalised variance bounds to

quantum

statistical estimation is shown to lead to higher order corrections to

the

Heisenberg uncertainty relations.

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