Chapter 1
Information Geometry of Smooth
Densities on the Gaussian Space:
Poincaré Inequalities
Giovanni Pistone
Abstract We derive bounds for the Orlicz norm of the deviation of a random variable
defined on R
n from its Gaussian mean value. The random variables are assumed to be
smooth, and the bound itself depends on the Orlicz norm of the gradient. We shortly
discuss possible applications to non-parametric Information Geometry.
Keywords Gaussian Poincaré-Wirtinger Inequality · Gaussian Space ·
Non-parametric Information Geometry · Orlicz Spaces
1.1 Introduction
In a series of papers [13, 24, 26, 27] we have explored a version of the non-parametric
Information Geometry (IG) for smooth densities on R
n . Especially, we have considered the IG associated to Orlicz spaces on the Gaussian space. The analysis of the
Gaussian space is discussed, for example, in [16, 21]. This set-up provides a simple
way to construct a statistical manifold modelled on Banach spaces of smooth densities. Other modelling options are in fact available, for example the global analysis
methods of [10], but we prefer to work with assumptions that allow for the use of
classical infinite dimensional differential geometry modelled on B-spaces as in [11].
The present note focuses on technical results about useful differential inequalities
and does not consider in detail the applications. However, we have in mind two main
examples of potential applications. The first one is the statistical estimation method
based on Hyvärinen’s divergence,
DH (P|Q) =
1
2
|∇ log P(x) − ∇ log Q(x)|
2 P(x) dx ,
(1.1)
The author is supported by de Castro Statistics, Collegio Carlo Alberto, Turin, Italy. He is a member
of GNAMPA-INDAM.
G. Pistone (B)
de Castro Statistics, Collegio Carlo Alberto, Piazza Arbarello 8, 10122 Torino, Italy
e-mail: giovanni.pistone@carloalberto.org
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_1
1
Information Geometry of Smooth
Densities on the Gaussian Space:
Poincaré Inequalities
Giovanni Pistone
Abstract We derive bounds for the Orlicz norm of the deviation of a random variable
defined on R
n from its Gaussian mean value. The random variables are assumed to be
smooth, and the bound itself depends on the Orlicz norm of the gradient. We shortly
discuss possible applications to non-parametric Information Geometry.
Keywords Gaussian Poincaré-Wirtinger Inequality · Gaussian Space ·
Non-parametric Information Geometry · Orlicz Spaces
1.1 Introduction
In a series of papers [13, 24, 26, 27] we have explored a version of the non-parametric
Information Geometry (IG) for smooth densities on R
n . Especially, we have considered the IG associated to Orlicz spaces on the Gaussian space. The analysis of the
Gaussian space is discussed, for example, in [16, 21]. This set-up provides a simple
way to construct a statistical manifold modelled on Banach spaces of smooth densities. Other modelling options are in fact available, for example the global analysis
methods of [10], but we prefer to work with assumptions that allow for the use of
classical infinite dimensional differential geometry modelled on B-spaces as in [11].
The present note focuses on technical results about useful differential inequalities
and does not consider in detail the applications. However, we have in mind two main
examples of potential applications. The first one is the statistical estimation method
based on Hyvärinen’s divergence,
DH (P|Q) =
1
2
|∇ log P(x) − ∇ log Q(x)|
2 P(x) dx ,
(1.1)
The author is supported by de Castro Statistics, Collegio Carlo Alberto, Turin, Italy. He is a member
of GNAMPA-INDAM.
G. Pistone (B)
de Castro Statistics, Collegio Carlo Alberto, Piazza Arbarello 8, 10122 Torino, Italy
e-mail: giovanni.pistone@carloalberto.org
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_1
1