Digital Signal Processing Reference
In-Depth Information
Section
4.4
is devoted to the twisted KV cohomology. The twisted KV complex
of
(
M
,
D
)
is a deformation of the KV complex of
(
M
,
D
)
. The classical De Rham
cohomology of a locally flat manifold
is canonically embedded in its twisted
KV cohomology. This viewpoint is efficient to discuss statistical structure when the
Fisher information is indefinite. Some results are obtained when the Fisher informa-
tions is the symmetric part of a twisted KV cocycle. In such a situation the statistical
geometry is a transverse geometry of a foliated manifold. Therefore, effective statis-
tical invariants are transverse (or bundlelike) invariants. By the same way the twisted
complex provides a relevant framework to discuss the so-called statistical structures
which are also called “Codazzi pair of order 2”. Actually, in a locally flat manifold
(
(
M
,
D
)
is nothing else that a
locally Hessian structure, [
27
]. If the model is compact then the locally flat structure
(
M
,
D
)
a statistical structure (or a Codazzi pair)
(
M
, g,
D
)
is hyperbolic, see Theorem 3 of [
16
].
We mainly deal with statistical models in the sense of [
1
]. Basic definitions are
stated. We relate the Fisher metric to random Hessian geometry. We use the Maurer
Cartan polynomial function of a KV complex to study the local convexity question
in the set of
α
-connections. Roughly speaking, let one assume to be dealing with the
vector space of linear connections. Rather than to study general deformations of a
given connection we restrict ourself to study deformations along a straight line
δ
of
connection. More clearly suppose you are interested in the set
M
,
D
)
∇
(
PR
)
of connections
with a fixed property
, (e.g. torsion free, flat, locally flat, symplectic and so on),
then move along a straight line
δ
in the vector space of linear connections and ask
how many plenty are the connections belonging to
PR
∇
(
P
)
∩
δ.
This question is called
PR
-convexity problem along
δ
. For instance let
δ
be a straight line through the Levi-
Civita connection of Fisher metric. Then arises the question to how many plenty are
the locally flat connections supported by
δ
. This question is briefly discussed as well.
Section
4.5
is devoted the duality in statistical models. Some comparison criteria
are discussed. Complete
α
-connections in two dimensional statistical structures are
discussed. We restrict ourself to complete orientable compact case as well as to the
2-dimensional gaussian model which nothing as the Poincaré half plane.
4.2 KV Complex and Statistical Structures
Some readers of this paper may have unfamiliarity with effective cohomological
algebra. Below is a simple straightforward colloquial introduction to the matter.
What the reader has to know the meaning of the following words and expressions.
Square of a linear endomorphism, kernel of a linear endomorphism, image of a
linear endomorphism, quotient of a vector space modulo a vector subspace.
(
,
)
1. A differential vector space is a couple
V
d
where
V
is a vector space and
d
is
a linear endomorphism of
V
such that
d
2
=
0.
The requirement
d
2
=
0 yields the inclusion relation
d
(
V
)
⊂
ker
(
d
)
.
Elements of
ker
(
d
)
are called (co)-cycles, those of
d
(
V
)
are called (co)
boundaries.
