102
F. Barbaresco
Geometry framework could be introduced by dual connections, and not only founded
on Fréchet, Rao and Chentsov works [103]. A hessian structure (D, g) is of Koszul
type, if there is a closed 1-form ω as g = Dω. Using D and the volume element of
g, Koszul introduced a 2nd form, which plays a similar role to the Ricci tensor for a
Kählerian metric. Let υ be the volume element of g, we define a closed 1-form α such
that D X υ = α(X )υ and a symmetric bilinear form γ = Dα. In the following, α and
γ forms are called 1st and 2nd form of Koszul for Hessian structure (D, g). We can
consider the forms associated with the Hessian dual structure (D
, g) by α
= −α
and γ
= γ − 2∇α. In the case of a homogeneous regular convex cone , with D
the canonical flat connection of the ambient vector space, the Koszul forms α and
γ for the canonical Hessian structure (D, g = Ddψ) are given by α = d log ψ and
γ = g. The volume element υ determined by g is invariant under the action of the
group of automorphisms G of .
Jean-Louis Koszul, invited by the intermediary of Professor Boyom, attended
the 1st GSI “Geometric Science of Information” conference in August 2013 at the
Ecole des Mines in Paris, where he attended the presentation of Hirohiko Shima,
given for his honor on the topic “Geometry of Hessian Structures “ [104]. In the
photo below, we can see from left to right, Jean-Louis Koszul and Hirohiko Shima.
Professor Michel Boyom has extensively studied and developed, at the University
of Montpellier, Koszul models [105, 106, 107, 108, 109, 85, 86, 87] in relation to
symplectic flat affine manifolds and to the cohomology of Koszul-Vinberg algebras
(KV Cohomology). Professor Boyom with his PhD student Byande [90, 110] have
explored other links with Information Geometry.
Links with Koszul-Vinberg characteristic function could be found in [111, 110]
(Fig. 5.3).
The main object studied by Information Geometry is invariant distance between
probability densities in the space of their parameters. The most natural metric has
been introduced by Rao to be the Fisher metric ds
2
θ that is invariant by changes of
non-singular parametrization:
w = W (θ ) ⇒ ds
2
w = ds
2
θ si en posant [I (θ )] i, j = −E
∂
2 log p θ (w)
∂θ i ∂θ j
ds
2
θ = −
p θ (w) log
p θ+dθ (w)
p θ (w)
dw
ds
2
θ ≈
T aylor
i, j
g i j dθ i dθ j =
i, j
[I (θ )] i, j dθ i dθ j = dθ
T
.I (θ ).dθ
linked with Fr ´
echet-Darmois-Cramer-Rao Bound:
E
θ − ˆ
θ
θ − ˆ
θ
T
≥ I (θ )
−1
(5.21)
For families of exponential densities, these parameters belong to homogeneous
bounded domains, we illustrate this property for Gaussian laws. For monovariate
F. Barbaresco
Geometry framework could be introduced by dual connections, and not only founded
on Fréchet, Rao and Chentsov works [103]. A hessian structure (D, g) is of Koszul
type, if there is a closed 1-form ω as g = Dω. Using D and the volume element of
g, Koszul introduced a 2nd form, which plays a similar role to the Ricci tensor for a
Kählerian metric. Let υ be the volume element of g, we define a closed 1-form α such
that D X υ = α(X )υ and a symmetric bilinear form γ = Dα. In the following, α and
γ forms are called 1st and 2nd form of Koszul for Hessian structure (D, g). We can
consider the forms associated with the Hessian dual structure (D
, g) by α
= −α
and γ
= γ − 2∇α. In the case of a homogeneous regular convex cone , with D
the canonical flat connection of the ambient vector space, the Koszul forms α and
γ for the canonical Hessian structure (D, g = Ddψ) are given by α = d log ψ and
γ = g. The volume element υ determined by g is invariant under the action of the
group of automorphisms G of .
Jean-Louis Koszul, invited by the intermediary of Professor Boyom, attended
the 1st GSI “Geometric Science of Information” conference in August 2013 at the
Ecole des Mines in Paris, where he attended the presentation of Hirohiko Shima,
given for his honor on the topic “Geometry of Hessian Structures “ [104]. In the
photo below, we can see from left to right, Jean-Louis Koszul and Hirohiko Shima.
Professor Michel Boyom has extensively studied and developed, at the University
of Montpellier, Koszul models [105, 106, 107, 108, 109, 85, 86, 87] in relation to
symplectic flat affine manifolds and to the cohomology of Koszul-Vinberg algebras
(KV Cohomology). Professor Boyom with his PhD student Byande [90, 110] have
explored other links with Information Geometry.
Links with Koszul-Vinberg characteristic function could be found in [111, 110]
(Fig. 5.3).
The main object studied by Information Geometry is invariant distance between
probability densities in the space of their parameters. The most natural metric has
been introduced by Rao to be the Fisher metric ds
2
θ that is invariant by changes of
non-singular parametrization:
w = W (θ ) ⇒ ds
2
w = ds
2
θ si en posant [I (θ )] i, j = −E
∂
2 log p θ (w)
∂θ i ∂θ j
ds
2
θ = −
p θ (w) log
p θ+dθ (w)
p θ (w)
dw
ds
2
θ ≈
T aylor
i, j
g i j dθ i dθ j =
i, j
[I (θ )] i, j dθ i dθ j = dθ
T
.I (θ ).dθ
linked with Fr ´
echet-Darmois-Cramer-Rao Bound:
E
θ − ˆ
θ
θ − ˆ
θ
T
≥ I (θ )
−1
(5.21)
For families of exponential densities, these parameters belong to homogeneous
bounded domains, we illustrate this property for Gaussian laws. For monovariate