Stein's method and Malliavin's calculus
organised by M. Clausel, J.F. Coeurjolly and J. Lelong
on the Poisson space
LJK, Grenoble University -- 24 June 2014
Laboratoire Jean Kuntzman, Tour IRMA, Room 1.
From the railway station, take Tram B (direction Gières - Plaine des
sports) and get off at the stop Bibliothèques Universitaires. The building
you are looking for is the highest one in your back. See here
for a map of the campus site.
Talks and schedule
- 11:30-12:15 Laurent Decreusefond,
Functional Poisson convergence via Stein method.
Consider a Poisson or a binomial process on some state space and a
symmetric function f acting on k--tuples of its points. This induces
a point process on the target space of f. If k=1, the image is again
a Poisson or a binomial process, a property which is not preserved for
k ≥ 2. We establish a functional limit theorem, providing an upper
bound for an optimal transportation distance between the image process
and a suitable Poisson process on the target space for general k. The
technical background are a version of Stein's method for Poisson process
approximation, a Glauber dynamic representation for the Poisson process
and the Malliavin formalism. The general theory is applied to
U--statistics as well as to geometric random graphs, random polytopes and
proximity functionals of non--intersecting flat processes.
- 12:15-14:00 Lunch
- 14:00-14:45 Dominic Schuhmacher,
Stein's method for Gibbs process approximation in the
total variation metric.
We consider Gibbs point processes on a compact metric space. Our goal is to
derive upper bounds for the total variation distance between a general
Gibbs process distribution and one that satisfies a certain stability
In order to achieve this, we adapt Stein's method to the Gibbs process
setting. We re--express the total variation distance in terms of the
generator of a spatial birth--death process and reduce the problem to
finding an upper bound of the expected coupling time between two spatial
birth--death processes with identical transition kernels but started at
different configurations. The later problem is solved by constructing an
explicit coupling and bounding the time until the two processes meet by
the hitting time at zero of a (non--spatial) birth--death process.
We obtain upper bounds of the form of an explicit constant times an
L1-type distance between the conditional intensities of the point
processes. As applications we consider the total variation distance
between two pairwise interaction processes and the hard core process
approximation of an area--interaction process with small interaction
- 14:45-15:30 Anthony Réveillac, Statistical inference for Poisson channels: a Malliavin calculus approach.
The aim of this talk is to present some statistical estimation results in
the context of infinite dimensional Poisson channels using the Malliavin
calculus. These channels differ from their Gaussian counterpart in
several aspects and Malliavin calculus brings, in that setting, new
perspectives. For example it allows one to derive a new proof of a
remarkable formula in Information Theory, relating the derivatives of the
input--output mutual information of a general Poisson channel and the
conditional mean estimator of the input, this holding regardless the
distribution of the latter. Furthermore, Malliavin calculus offers a
universal framework for studying a quite general class of channels.
Finally, we will discuss some open questions in that setting.
- 15:30-:16:00 Coffee break
- 16:00-16:45 Raphaël Lachièze-Rey, Different scaling regimes for Poisson U--statistics.
Recent results combining Malliavin calculus and Stein's method triggered
a series of limit theorems for geometric functionals on the Poisson
space. The applications include statistics on the boolean model,
intersections of flat processes, estimation of the Sylvester constant,
subgraph counting, or the Voronoi approximation of a set, with Gaussian,
Poisson, or Gamma limit. We will present some of these results for the
functionals that can be put under the form of a Poisson U--statistics, or
equivalently for finite sums of multiple integrals with respect to the
Poisson measure. It appears that a crucial feature is the area of
interaction of the U--statistics, i.e. the average number of points
interacting with a typical point. It appears that in many cases the limit
law and speed of convergence magnitude depend on this quantity,
characterising the scaling regime of the functional.
- 16:45-17:30 Jean-François Coeurjolly, Stein's estimation of the intensity of a stationary spatial Poisson point process.
We revisit the problem of estimating the intensity parameter of a
homogeneous Poisson point process observed in a bounded window of Rd making use of a (now) old idea of James
and Stein. For this, we prove an integration by parts formula for
functionals defined on the Poisson space. This formula extends the one
obtained by Privault and Réveillac (Statistical inference for Stochastic
Processes, 2009) in the one-dimensional case. As in Privault and Réveillac,
this formula is adapted to a notion of gradient of a Poisson functional
satisfying the chain rule, which is the key ingredient to propose new
estimators able to outperform the maximum likelihood estimator (MLE) in
terms of the mean squared error.
The new estimators can be viewed as biased versions of the MLE but with a
well--constructed bias, which reduces the variance. We study a large class
of examples and show that with a controlled probability the corresponding
estimator outperforms the MLE. We will illustrate in a simulation study
that for very reasonable practical cases (like an intensity of 10 or 20 of
a Poisson point process observed in the euclidean ball of dimension
between 1 and 5) we can obtain a relative (mean squared error) gain of
20% of the Stein estimator with respect to the maximum likelihood.
Registration and participants
Attending the workshop is free of charge but registration is compulsory.
- Gilles Bonnet (University of Osnabrueck)
- Sébastien Darses (Unisersité Aix-Marseille)
- Laurent Decreusefond (Telecom ParisTech)
- Vincent Douchez (INRIA Grenoble)
- Rémy Drouihlet (Université de Grenoble)
- Pierre Etoré (Université de Grenoble)
- Gérard Grégoire (Université de Grenoble)
- Céline Labart (Université de Savoie)
- Raphaël Lachièze-Rey (Université Paris Descartes)
- Ester Marriuci (Université de Grenoble)
- Nadia Morsli (Université de Grenoble)
- Anthony Réveillac (Université Paris Dauphine)
- Raphaël Rossignol (Université de Grenoble)
- Dominic Schuhmacher (Georg-August-Universität Göttingen)
We thank our partners for their financial support.