Stephen P. King (stephenk1@home.com)
Mon, 21 Jun 1999 20:23:32 -0400
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
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