Stephen P. King (stephenk1@home.com)
Fri, 11 Jun 1999 19:55:05 -0400
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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