5 Invariant Koszul Form of Homogeneous Bounded Domains …
103
Fig. 5.3 Jean-Louis Koszul and Hirihiko Shima at GSI’13 “Geometric Science of Information”
conference in Ecole des Mines ParisTech in Paris, October 2013
gaussian density, parameters are given by θ =
m
σ
with m the mean and σ the
standard deviation of p θ (w) =
1
√
2πσ 2 e
−
1
2
(w−m) 2
σ 2
. Fisher metric computation provides:
I (θ ) =
1
σ 2 0
0
2
σ 2
⇒ ds
2
θ = dθ
T
.I (θ ).dθ =
dm
2
σ 2 + 2.
dσ
2
σ 2
ds
2
θ = 2.
|dz|
2
(Im(z))
2
with z =
m
√
2
+ iσ
(5.22)
We recover the metric of the Poincaré upper-half plan H = {z = x + iy/y > 0},
the most simple example of homogeneous bounded domain. It is obvious to recover
this geometry for the Fisher metric of Gaussian densities, because their space of
parameters θ =
m
σ
lie in H Gaussienne =
z = m + i
√
2σ/m, σ ∈ R, σ > 0
,
the upper half plan. Fisher metric invariance by reparametrization inherits this property by a more richer invariance given by invariance by automorphisms of this
homogeneous bounded domain. Poincaré metric ds
2
=
dx
2
y 2 +
dy
2
y 2 =
|dz|
2
(Im(z))
2 =
y
−1 dzy
−1 dz
∗ for upper-half plan is invariant by its automorphisms given by M(z) =
az+b
c.z+d
with ad − bc = 1. Then, a Gaussian density is coded by 1 point in the upper