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

*Mon, 21 Jun 1999 20:23:32 -0400*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 415] Re: On the Problem of Information Flow between LSs"**Previous message:**Stephen P. King: "[time 413] Re: Symplectic Geometry and GR"

Dear Hitoshi and Matti,

I am recommending Shun-ichi Amari's Differential-Geometrical Methods in

Statistics, Springer-Verlag 1985 to work out the problem of defining the

causal space-times generated by LS interactions.

My Reasoning:

"Statistical inference can be carried out more and more precisely as the

number of observations increases, so that one can construct a universal

asymptotic theory of statistical inference in the regular case. Since

the estimated probability distribution lies very close to the true

distribution in this case, it is sufficient when evaluating statistical

procedures to take into account of only the local structure of the model

in a small neighborhood of the true or estimated distribution. Hence,

one can locally linearize the model at the true or estimated

distribution, even if the model is curved in the entire set.

Geometrically, this local linearization is an approximation to the

manifold by the tangent space at a point. The tangent space has a

natural space has a natural inner produced (Riemannian metric) given by

the Fisher information matrix. From the geometrical point of view, one

may say that the asymptotic theory of statistical inference has indeed

been constructed by using the linear geometry of tangent spaces of a

statistical model, even if it has not been explicitly stated.

Local linearization accounts only for local properties of a model. In

order to elucidate larger-scale properties of a model, one needs to

introduce mutual relations of two different tangent spaces at two

neighboring points in the model. This can be done by defining an affine

correspondence between two tangent spaces at neighboring points. This is

a standard technique of differential geometry and the correspondence is

called an affine connection. By an affine connection, one can study

local non-linear properties such as curvature, of a model beyond linear

approximations. This suggests that a a higher-order asymptotic theory

can naturally be constructed in the framework of differential geometry.

Moreover, one can obtain global properties of a model by connecting

tangent spaces at various points." pg. 1-2, ibid.

Onward to the Unknown,

Stephen

**Next message:**Matti Pitkanen: "[time 415] Re: On the Problem of Information Flow between LSs"**Previous message:**Stephen P. King: "[time 413] Re: Symplectic Geometry and GR"

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