2
G. Pistone
where |·| denotes the Euclidean norm of R
n , P, Q are positive probability densities
of the n-dimensional Lebesgue space, see in [8, 13]. The second one is the Otto’s
inner product [14, 22], which is defined by
f, g P =
∇ f (x) · ∇g(x) P(x) dx ,
(1.2)
where P is a probability density and f, g are smooth random variables such that
E P [ f ] = E P [g] = 0.
We focus on the exponential representation of positive densities P = p · γ =
e
u−K (u)
· γ , where γ is the standard Gaussian density. The sufficient statistics u is
assumed to belong to an exponential Orlicz space and
u(x) γ (x) dx = 0. The set of
all such couples ( p, u) is called statistical bundle. There are other ways to represent
positive densities, namely, those that use deformed exponential functions, p ∝ exp A
[18]. This approach is intended to avoid the difficulty of the exponential growth and,
for this reason, provides a somehow simpler treatment of smoothness, see [19, 20].
We do not further discuss here this interesting formalism.
This paper is organized as follows. In Sect. 1.2, we provide a recap of basic
facts about non-parametric IG and introduce the Gaussian case. The results about
Poincaré-Wirtinger inequalities are gathered in Sect. 1.3. This section contains the
main contributions of the paper. A collection of simple examples of possible applications concludes the paper.
1.2 Statistical Bundle Modelled on Orlicz Spaces
First, we review below the theory of Orlicz spaces in order to fix convenient notation.
The full theory is offered, for example, in [17, Chap. II] and [1, Chap. VII].
1.2.1 Orlicz Spaces
In this paper, we will need the following special type of Young function.
Cf. [17, Sect. 7] for a more general case.
Assume φ ∈ C[0, +∞[ is such that: (1) φ(0) = 0; (2) φ(u) is strictly increasing;
(3) lim u→+∞ φ(u) = +∞. The primitive function
Φ(x) =
x
0
φ(u) du , x ≥ 0 ,
is strictly convex and will be called a Young function. Cf. [1, Sect. 8.2], where φ is
assumed to be right-continous and non-decreasing.
G. Pistone
where |·| denotes the Euclidean norm of R
n , P, Q are positive probability densities
of the n-dimensional Lebesgue space, see in [8, 13]. The second one is the Otto’s
inner product [14, 22], which is defined by
f, g P =
∇ f (x) · ∇g(x) P(x) dx ,
(1.2)
where P is a probability density and f, g are smooth random variables such that
E P [ f ] = E P [g] = 0.
We focus on the exponential representation of positive densities P = p · γ =
e
u−K (u)
· γ , where γ is the standard Gaussian density. The sufficient statistics u is
assumed to belong to an exponential Orlicz space and
u(x) γ (x) dx = 0. The set of
all such couples ( p, u) is called statistical bundle. There are other ways to represent
positive densities, namely, those that use deformed exponential functions, p ∝ exp A
[18]. This approach is intended to avoid the difficulty of the exponential growth and,
for this reason, provides a somehow simpler treatment of smoothness, see [19, 20].
We do not further discuss here this interesting formalism.
This paper is organized as follows. In Sect. 1.2, we provide a recap of basic
facts about non-parametric IG and introduce the Gaussian case. The results about
Poincaré-Wirtinger inequalities are gathered in Sect. 1.3. This section contains the
main contributions of the paper. A collection of simple examples of possible applications concludes the paper.
1.2 Statistical Bundle Modelled on Orlicz Spaces
First, we review below the theory of Orlicz spaces in order to fix convenient notation.
The full theory is offered, for example, in [17, Chap. II] and [1, Chap. VII].
1.2.1 Orlicz Spaces
In this paper, we will need the following special type of Young function.
Cf. [17, Sect. 7] for a more general case.
Assume φ ∈ C[0, +∞[ is such that: (1) φ(0) = 0; (2) φ(u) is strictly increasing;
(3) lim u→+∞ φ(u) = +∞. The primitive function
Φ(x) =
x
0
φ(u) du , x ≥ 0 ,
is strictly convex and will be called a Young function. Cf. [1, Sect. 8.2], where φ is
assumed to be right-continous and non-decreasing.