2012 - 2013
Organisateurs : Nicolas CHOPIN
et Gersende FORT
- Jeudi
13 Juin [Ampĥi
Perrin]
- Jeudi
16 Mai [Amphi
Hermite]
- [15h]
Pierre Del Moral (INRIA
Bordeaux)
- Titre :
Particle
approximation of multiple object nonlinear filtering
problems
- Résumé
: We
consider the problem of estimating a latent point
process, given the realization of another point
process on abstract measurable state spaces.
First, we
establish an expression of the conditional
distribution of a latent Poisson point process given
the observation process when the transformation from
the latent process to the observed process includes
displacement, thinning and augmentation with extra
points. We present an original analysis based on a
self-contained random measure theoretic approach
combined with reversed Markov kernel techniques. In
the second part, we analyse the exponential
stability properties of nonlinear multi-target
filtering equations. We prove uniform convergence
properties w.r.t. the time parameter of a rather
general class of stochastic filtering algorithms,
including sequential Monte Carlo type models and
mean field particle interpretation models. We
illustrate these results in the context of the
Bernoulli and the Probability Hypothesis Density
filter.
- [16h15] François
Septier
(Telecom
Lille 1)
- Titre
: Bayesian Filtering in
High-Dimensional
Spaces using Sequential MCMC
- Résumé
: Nonlinear non-Gaussian
state-space
models arise in numerous
applications in control and signal processing. In
this context,
one of
the most successful and popular
approximation techniques is
Sequential
Monte Carlo (SMC) methods, also known
as particle filters.
Nevertheless, these methods tend to be
inefficient when applied to
high
dimensional problems. In this talk, I will
present an overview of
Markov chain Monte Carlo (MCMC) methods for
sequential simulation
from
posterior distributions, which represent efficient
alternatives to
SMC methods especially in high dimensional
spaces. Then, I will
describe ongoing work on Sequential
Markov chain Monte Carlo
(SMCMC)
algorithms in which we incorporate a component of
Approximate
Bayesian
Computation (ABC) in order to obtain a sequential
estimation
methodology for a class of non-linear, non-Gaussian
state space models
in which the observation process is intractable
to express in
closed
form. Numerical simulations applied to multiple
target tracking
and
sensor network will be presented.
- Jeudi
18 Avril [Amphi
Hermite] (séance
annulée)
- Jeudi
28 Mars
[Salle 314]
- [15h]
Yves Atchadé (Univ. Michigan, USA)
- Titre: Assessing Monte Carlo
errors in MCMC and
adaptive MCMC
-
Résumé:
The
talk
will
present
some asymptotic results on quadratic forms of Markov
chains and adaptive Markov chains. The results are
used to derive Monte
Carlo confidence intervals using the standard
Gaussian distribution,
and the so-called fixed-b asymptotic. We will also
discuss some
convergence rates of quadratic forms that give some
insight on the
rates of convergence of these confidence intervals.
- [16h15] O.
Papaspiliopoulos
(UPF, Espagne)
- Titre:
Optimal
filtering and
the dual process
- Résumé:
In
this
talk I will present ongoing work on a class of
partially observed
Markov models for which the so-called filtering
distributions belong in
mixtures of known finite dimensional distributions,
the parameters and
weights of which can be computed sequentially. This
class contains as
special case the models for which the well-known Baum
or Kalman filters
can be used for the computation of the filtering
distributions, and
involves high and infinite-dimensional unobserved
signals. The filters
for this class are computable because the law of the
unobserved signals
in these models admits a representation in terms of an
auxiliary
discrete state-space, continuous-time Markov chain.
This auxiliary
process is directly related with the so-called dual
process in
population genetics, and as we show the processes in
the class we
identify are related to the so-called Fleming-Viot
process.
- Joint work with
Matteo
Ruggiero (University of Turin)
- Jeudi
21 Février
[Salle 314]
- [15h]
Benjamin Jourdain
et Tony Lelièvre
(ENPC)
- Titre
: Optimal scaling of the
transient phase of
Metropolis Hastings algorithms
- Résumé
: We
consider
the
Random
Walk Metropolis algorithm on R^n with Gaussian
proposals,
and when the target probability measure is the n-fold
product of a one
dimensional law. It is well-known that, in the limit n
tends to
infinity, starting at equilibrium and for an
appropriate scaling of the
variance and of the timescale as a function of the
dimension n, a
diffusive limit is obtained for each component of the
Markov chain. We
generalize this result when the initial distribution
is not the target
probability measure. The obtained diffusive limit is
the solution to a
stochastic differential equation nonlinear in the
sense of McKean. We
prove convergence to equilibrium for this equation. We
discuss
practical counterparts in order to optimize the
variance of the
proposal distribution to accelerate convergence to
equilibrium. Our
analysis confirms the interest of the constant
acceptance rate strategy
(with acceptance rate between 1/4 and 1/3).
- [16h15] Adam M.
Johansen (Univ.
Warwick,
UK)
- Titre
: Exact Approximation of Rao-Blackwellised
Particle Filters
- Résumé
: Particle
methods
are a category of Monte Carlo algorithms that have
become
popular for performing inference in non-linear
non-Gaussian state-space
models. The class of “Rao- Blackwellised” particle
filters exploits the
analytic marginalisation that is possible for some
state- space models
to reduce the variance of the Monte Carlo estimates.
Despite being
applicable to only a restricted class of state-space
models, such as
conditionally linear Gaussian models, these algorithms
have found
numerous applications. In scenarios where no such
analytical
integration is possible, it has recently been proposed
in Chen et al.
[2011] to use “local” particle filters to carry out
this integration
numerically. We propose here an alternative approach
also relying on
“local” particle filters which is more broadly
applicable and has
attractive theoretical properties. Proof-of-concept
simulation results
are presented.
- Joint work with
Nick
Whiteley and Arnaud Doucet
- T. Chen, T.
Schoen, H.
Ohlsson, and L. Ljung. Decentralized particle filter
with arbitrary
state decomposition. IEEE Trans. Sig. Proc.,
59(2):465–478, 2011
- Jeudi
10 Janvier 2013
[Salle 314] (séance
annulée)
- Jeudi
15 Novembre [Salle
201]
- [15
h] Jean-Michel
Marin (Univ.
Montpellier 2)
- Titre
: Consistency
of
Adaptive Multiple Importance Sampling
- Résumé
:Among Monte
Carlo techniques,
the importance sampling requires fine tuning
of a proposal
distribution, which is now fluently done with
iterative schemes.
The Adaptive
Multiple Importance Sampling (AMIS) of Cornuet et al.
(2012) provides a
significant improvement in stability and Effective Sample
Size due to the
introduction of a recycling procedure. However, the
consistency of the AMIS
estimator remains largely open. In this work we
provides proofs of the
convergence of the AMIS, at a cost of a slight modification
in the learning
process. First numerical experiments exhibit that
this modification might
even improve the original scheme.
- Réf. Cornuet,
J.-M., Marin, J.-M.,
Mira, A., and Robert, C. P. (2012). Adaptive
Multiple Importance
Sampling.
Scandinavian Journal of Statistics.
- [16h15]
Benjamin Guedj (LSTA,
UPMC &
LTCI, Telecom ParisTech)
- Titre
: A Stochastic Search
MCMC algorithm for
sparse additive models.
- Résumé
: Penalty-based
estimators
such
as
the
Lasso proved successful in addressing high-dimensional
regression problems under a constraint of sparsity.
However, their good
theoretical properties only hold under stringent
conditions on the
design (mutual coherence, R.I.P.), which may appear
too much of a
burden when it comes to building prediction
algorithms. In this work,
an alternative PAC-Bayesian strategy in investigated,
carrying no
assumption on the design. We
will focus on the explicit implementation of our
exponentially weighted
estimator. Our procedure relies on a stochastic search
throughout the
models space, the key idea being to favor local moves
of the Markov
Chain. Numerical evidence of the relevance of our
method will be
provided on simulated data.
- Réf.
Joint work with Éric Moulines (LTCI, Telecom
ParisTech),
Gérard Biau (LSTA, UPMC & IUF), Pierre
Alquier (School of
Mathematical Sciences, University College of Dublin).http://arxiv.org/abs/1208.1211
http://www.cran.r-project.org/web/packages/pacbpred/index.html
- Jeudi
18 Octobre [Amphi
Hermite]
- [15
h] Orateur : Olivier
Ratmann
(Univ.
Duke, USA)
- Titre :
Feature-based
inference
of
virus
phylodynamics, with
ABC
calibrated for exact parameter inference
- Résumé
: The
infectious
disease
dynamics
of
many viral pathogens like influenza,
norovirus and
coro- navirus are inextricably tied to their evolution.
This interaction
between evolutionary and ecological processes
complicates our ability to understand the
infectious disease behavior
of
rapidly evolving pathogens. Most statistical methods
for the
analysis of these
“phylodynamics” require that the likelihood of the data
can be explicitly
calculated. Currently, this is not possible for many
phylodynamic models,
so that questions on the interaction between
viral variants cannot
be well-addressed within this framework. Simulation-based
statistical
methods circumvent likelihood calculations.
I
here illustrate the effectiveness of these methods to fit and
assess complex
phylodynamic models against both sequence and surveillance
data, and also
present a new, fairly broadly applicable, theoretical
framework for
standard ABC parameter inference that aims to improve
on existing,
rather heuristic algorithm formulations.
- [16 h15]
Orateur
: Rémi Bardenet (LAL
& LRI, Univ Orsay)
- Titre
:
When cosmic particles switch labels: Adaptive
Metropolis with online relabeling, motivated by the
data analysis of
the Pierre Auger experiment
- Résumé
:
The Pierre Auger experiment is a giant cosmic ray
observatory located
in Argentina. Cosmic rays are charged particles that
travel through the
universe at very high energies. When one of these
particles hits our
atmosphere, it generates a cascade of particles that
strike the surface
of Earth on several square kilometers. Auger has a
3000 km2 wide array
of 1600+ detectors gathering data from these cascades.
The objective of
the data analysis is to infer the parameters of the
original incoming
particles. In this talk, we first derive a model of
part of the
detection process, which involves elementary particles
called muons.
The resulting model is invariant to permutations of
these muons, thus
making MCMC inference prone to label-switching,
similarly to what
happens with MCMC in mixture models. In addition, our
model is high
dimensional and involves a lot of correlated
variables, which motivates
the use of adaptive MCMC, such as the adaptive
Metropolis (AM) of
Haario et al. (Bernoulli, 2001). However, running AM
on our model
requires to solve the label-switching online. Building
on previous
approaches, we present AMOR, a variant of AM that
learns together an
optimal proposal and an optimal relabeling of the
marginal chains. We
present applications and state convergence results for
AMOR, including
a law of large numbers, and demonstrate interesting
links between
relabeling and vector quantization.
- Jeudi
20 Septembre [Amphi
Darboux]
- [15
h] Orateur : Kengo
Kamatani
(Graduate
School
of
Engineering
Science,
Japan)
- Titre
: Local consistency of
MCMC and its
application to cumulative link model
- Résumé
: Cumulative
link
model
is
a
general model for ordinal data which
includes binary probit model as a special case. Bayesian
inference for the
model is usually performed by a
simple
data augmentation strategy, which is a kind of
Markov chain Monte
Carlo (MCMC). However,
It
is known
to work poorly
except some simple cases. Two
MCMC
strategies are described in this talk. One is a
simple extension of
marginal augmentation strategy and the other
is a kind of multi-step MCMC. The efficiency is discussed
in terms of local consistency, which
is
a large sample asymptotic property. We discuss
some properties of
local consistency and its other applications.
2011 - 2012
- Jeudi 14 Juin [Amphi
Hermite]
- [15h] Orateur : Cyrille
Dubarry (Telecom SudParis)
- Titre : On
the
convergence of Island particle models
-
Résumé :
Numerical approximation of Feynman-Kac semigroups by
systems of
interacting particles is a very active research
field. Such methods are
increasingly used to sample complex high dimensional
distributions and
they found a wide range of applications in applied
probability, among
others filtering, smoothing for non linear
state-space models, Bayesian
inference of hierarchical models, branching
processes in biology,
absorption problems in physics.
The
sequences
of distributions involved are approximated
sequentially using
interacting particle systems. Such particle
approximations are often
referred to as sequential Monte-Carlo (SMC) methods.
The asymptotic
behavior of such particle approximation is now well
understood.
Recently,
the
development of parallel computing methods lead to
the study of
parallel implementation of this particle
approximation. In this paper
we focus on the so-called island particle models,
consisting in running
N2 interacting particle systems each of size N1.
This problem gives
rise to several questions among which the
optimization of the size of
the islands N1 compared to the number of islands N2
for a given total
computational cost and the opportunity of letting
the islands interact.
When
the
N2 islands are run independently, the bias induced
in each of them
only depends on their population size N1; thus it
can be interesting to
introduce an interaction between the islands even
though this
interaction increases the final estimator variance
through the sampling
steps. We propose here to study the asymptotic bias
and variance in
each case so that when N1 and N2 are large and
fixed, we can determine
which choice is the best in terms of asymptotic mean
squared error.
Keywords: Feynman-Kac
model,
Particle
approximation,
Island
particle model, Asymptotic variance, Asymptotic
bias
Joint work with : P. Del Moral (ALEA, INRIA
Bordeaux), E. Moulines (LTCI, Telecom ParisTech).
- [16 h] Orateur : Julien
Cornebise (University
College
London)
- Titre
: MCMC Particulaire Adaptatif pour Modèles
à Effets
Mixtes Stochastiques
- Résumé
: MCMC
Particulaires et MCMC Adaptatives sont une combinaison
puissante pour
des modèles à structure complexe; nous
présentons
ici notre expérience sur des modèles
non-linéaires
à effets-mixtes, basés sur des
équations
différentielles stochastiques, pour des
études
longitudinales sur population. Nous expliquons
pourquoi la
complexité de ces modèles
pharmacocinétique/pharmacodynamique, tant au
niveau
stochastique, que temporel et hiérarchique en
fait des terrains
de jeux fabuleux pour la statistique
computationnelle.
L'application d'intérêt est un
système de
régulation glucose/insuline sur une cohorte de
patients; par
ailleurs, la même structure de dépendance
se retrouve bien
au-delà de la pharmacologie: agronomie,
retraitement des eaux,
sylviculture...
Nous montrons ensuite comment les méthodes de
Monte-Carlo
adaptatives peuvent fortement améliorer la
mélangeance du
MCMC Particulaire; en particulier nous exploitons
certaines relations
d'indépendance conditionnelle pour
réduire
considérablement le nombre de paramètres
à
adapter; nous montrons également que ceci
permet dans de
nombreux cas une proposition exacte des
paramètres de
population, alternant étapes de Particle
Metropolis Hastings et
d'échantillonneur de Gibbs. Enfin, nous
esquissons des liens
avec les extensions récentes de la
communauté MCMC pour
incorporer les aspects géométriques du
modèle en
étendant les approches Riemannian Manifold MALA
aux
chaînes de Markov cachées.
- Travaux en cours avec Arnaud Doucet et Gareth
W. Peters
- Jeudi 10 Mai [Amphi
Hermite] Attention,
changement d'horaire
(exceptionnellement)
- [15h30] Orateur : Meili Baragatti
(MISTEA,
SupAgro)
- Titre
: Parallel
Tempering with
Equi-Energy Moves, and Likelihood Free Parallel
tempering
-
Résumé
: We will consider the
common problem of generating samples from a target
density $\pi$, in
particular using Population-based Monte Carlo markov
Chain approaches. In conventional MCMC methods
(Metropolis-Hastings, Gibbs Sampler), a Markov
process is built to
sample the target probability distribution. But in
practice this process
can be easily trapped into a local mode from where
it cannot escape in
reasonable time. Many
techniques have been proposed to address this
waiting time problem,
including among others Parallel Tempering (PT) (see
Geyer [1991] or
Geyer and Thompson [1995]), and the Equi-Energy
Sampler (EES) (Kou et
al. [2006]). The EES is based on a population of
chains which are
updated by local moves and global moves, also called
Equi-Energy jumps.
The state space is partitioned into energy rings,
and the current state
of a chain can jump to a past state of an adjacent
chain that has an
energy level close to its level. This algorithm has
been developed to
facilitate global moves between different chains,
resulting in a good
exploration of the state space by the target chain. This
method seems to be more
efficient than the classical PT algorithm. However
the process obtained
is non Markovian, it is difficult to use in
combination with a Gibbs
sampler and it necessitates increased storage. We will
present an adaptation
of this EES that combines PT with the principle of
swapping between
chains with same levels of energy. This adaptation,
that we shall call
Parallel Tempering with Equi-Energy Moves (PTEEM),
keeps the original
idea of the EES method while generating a Markovian
process and
requiring less storage. Performances of the PTEEM
algorithm will be
compared with those of the EES and of the standard
PT algorithms in the
context of mixture models, and in a problem of
identification of
transcription factor binding motifs.
In a second
part of the
presentation, we will present an other Population-based
Monte Carlo
markov Chain approach which can be use in an Approximate
Bayesian
Computational (ABC) framework (or likelihood-free
framework). ABC
methods have appeared in the past fifteen years as useful
methods to
perform Bayesian analysis when the likelihood is
analytically or
computationally intractable. Several approaches have been
proposed:
Monte Carlo Markov Chains (MCMC) methods have been developed
by
Marjoram et al. [2003] and by Bortot et l. [2007] for
instance, and
sequential methods have been proposed among others by Sisson
et al.
[2007], Beaumont et al. [2009] and Del Moral et al. [2012].
Until
recently, while ABC-MCMC methods were the reference,
sequential ABC
methods have appeared to outperform them (see for example
McKinley et
al. [2009] or Sisson et al. [2007]). We will present a new
algorithm
combining population-based MCMC methods with ABC
requirements, using an
analogy with the Parallel Tempering algorithm.
Joint work with A. Grimaud, D. Pommeret.
- Références :
- Baragatti M, Grimaud A, Pommeret D. Likelihood-Free
Parallel
Tempering, Statistics and Computing, in press, 2012.
- Baragatti M, Grimaud A, Pommeret D. Parallel
tempering with
Equi-Energy moves, Statistics and Computing, in
press, 2012.
- [16h40] Orateur
: Kerrie
Mengersen
(Queensland Univ. of Technology , Australia)
- Titre : Understanding
images:
from
inferential
aims
to models to algorithms
- Résumé
: In
the
excitement of working with algorithms, it is sometimes
salutory to remind
ourselves of their
purpose. In this presentation, we consider the analysis of
image data and
try to match inferential aims, models and computational
methods. We
describe and compare the approaches in the context of
some real case
studies in agriculture and environmental monitoring.
- Jeudi 12 Avril [Amphi
Darboux]
- [15h] Orateur : Pierre
Jacob
(ENSAE) et Robin
Ryder (CEREMADE)
- Titre : Some
aspects
of
the
Wang-Landau algorithm.
- Résumé
: The
Wang-Landau algorithm is an adaptive MCMC algorithm
which generates a
Markov chain designed to move efficiently in the state
space, by
constantly penalizing already-visited regions. It
hence falls into the
class of exploratory algorithms, especially when the
chosen regions
correspond to different levels of density values. We
explore two novel
aspects of the Wang-Landau algorithm. First, we show
that the algorithm
reaches the so-called Flat Histogram criterion in
finite time, which
ensures convergence properties. Second, we examine the
effect of using
multiple chains, interacting through a common
component. That component
essentially represents the history of already-visited
regions, computed
on all the chains. We show numerically the benefit of
using parallel
chains even if a single processing unit is available,
in terms of
stabilization of the schedule used in the adaptation
process. If time
permits, we shall present an ongoing attempt to study
theoretically the
effect of parallelization using Feynman-Kac
semigroups.
- Références http://arxiv.org/abs/1110.4025
et http://arxiv.org/abs/1109.3829
- [16h] Orateur
: Nick
Whiteley ( Univ.
Bristol, UK)
- Titre
:
A particle method for approximating principal
eigen-functions and
related quantities
- Résumé :
Perron-Frobenius theory treats the existence of a
positive eigen-vector
associated with the principal eigen-value
\lambda_{\star} of a
non-negative matrix, say Q. A simple method for
approximating this
eigen-vector involves computing the iterate
\lambda_{\star}^{-n}Q^{(n)}, for large n. In the more
general case that
Q is a non-negative integral kernel, an extended
Perron-Frobenius
theory applies, but it is typical that neither the
principal
eigen-function nor the iterate
\lambda_{\star}^{-n}Q^{(n)} can be
computed exactly. In this setting we introduce an
interacting particle
algorithm which yields a numerical approximation of
the principal
eigen-function and the associated twisted Markov
kernel. Some of its
theoretical properties will be discussed and
applications will be
outlined. In particular, the algorithm allows
approximation of an
optimal importance sampling method for Markov chain
rare event
estimation.
Joint work with Nikolas Kantas.
- Référence
: http://arxiv.org/abs/1202.6678
- Jeudi 15 Mars [salle
314]
- Jeudi 9
Février
[Amphi Darboux]
- [15h] Orateur : A.
Kebaier
(LAGA, Paris 13)
- Titre :
Central
limit
theorem
for
the
multi-level algorithm
- Résumé
: This paper
deals with the problem of the multi-level Monte Carlo
method, introduced
by Giles
(Multilevel Monte Carlo path simulation Operations Research,
2008; 56:607-617)
as an extended method of the statistical Romberg one
of Kebaier
(Romberg Extrapolation: A New Variance Reduction Method and
Applications to
Option Pricing. Ann. Appl. Probab. 15 (2005), no. 4,
2681-2705). When
approximating the expected value of a function of a
stochastic differential
equation solution, these methods improve efficiently
the computational
complexity of standard Monte Carlo. In this work,
we analyze the asymptotic
error of this
algorithm and establish a central limit theorem
based on a new stable
functional central
limit
theorem on the error in the Euler scheme for a given level. This
allows us to
obtain the optimal choice of the parameters method. Then,
we
investigate the application of this method to the
pricing of Asian
options. In this setting, the
approximation relies on the discretization of the
integral of the price
process over a time interval.
We
also analyze the error process and prove a stable
functional central
limit theorem.
Finally, We use our result in order to optimize the choice of the
parameters, which are
different from the ones in the Euler scheme. Numerical
simulations were
processed.
- [16h15] Orateur
: N. Chopin
(CREST)
- Titre : SMC2:
A
sequential Monte Carlo algorithm with particle Markov
chain Monte Carlo
updates
-
Résumé :
We consider
the generic problem of performing sequential
Bayesian inference in a
state-space model with observation process y, state
process x and fixed
parameter theta. An idealized approach would be to
apply the iterated
batch importance sampling (IBIS) algorithm of Chopin
(2002). This is a
sequential Monte Carlo algorithm in the
theta-dimension, that samples
values of theta, reweights iteratively these values
using the
likelihood increments p(y_t|y_1:t-1, theta), and
rejuvenates the
theta-particles through a resampling step and a MCMC
update step. In
state-space models these likelihood increments are
intractable in most
cases, but they may be unbiasedly estimated by a
particle filter in the
x-dimension, for any fixed theta. This motivates the
SMC^2 algorithm
proposed in this article: a sequential Monte Carlo
algorithm, defined
in the theta-dimension, which propagates and
resamples many particle
filters in the x-dimension. The filters in the
x-dimension are an
example of the random weight particle filter as in
Fearnhead et al.
(2010). On the other hand, the particle Markov chain
Monte Carlo
(PMCMC) framework developed in Andrieu et al. (2010)
allows us to
design appropriate MCMC rejuvenation steps. Thus,
the theta-particles
target the correct posterior distribution at each
iteration t, despite
the intractability of the likelihood increments. We
explore the
applicability of our algorithm in both sequential
and non-sequential
applications and consider various degrees of
freedom, as for example
increasing dynamically the number of x-particles. We
contrast our
approach to various competing methods, both
conceptually and
empirically through a detailed simulation study,
included here and in a
supplement, and based on particularly challenging
examples.
Joint work with Pierre E. Jacob, and
Omiros
Papaspiliopoulos.
-
- Jeudi 12 Janvier
[Amphi
Darboux]
- [15h] Orateur :
S.
Allassonnière
(CMAP, Ecole Polytechnique)
- Titre :
Anisotropic
Metropolis
Adjusted Langevin Algorithm: convergence and utility
in
Stochastic EM algorithm
- Résumé
: Sampling a
random variable in a high dimensional framework
becomes a crucial issue
regarding the large range of applications which
require such samples,
such as image analysis. The Langevin simulation method
is well adapted
in this context. However, the use of an isotropic
covariance matrix
usually prevents from a numerical convergence which
suffers from the
low acceptation rate. In this work, we propose to
adapt the Langevin
simulation taking into account the anisotropy of the
distribution to
sample from. We prove that this new algorithm leads to
a uniform
ergodic Markov chain. Moreover, we apply
this new Monte
Carlo Markov Chain method into a stochastic
Expectation-Maximization
algorithm in order to estimate some model parameter by
maximizing the
likelihood. We prove that under mild conditions, the
estimated
parameters converge almost surely and are
asymptotically Gaussian
distributed. All this pipeline is numerically tested
on handwritten
digits and some medical images for the deformable
template estimation.
- [16h15] Orateur
: A.
Fulop
(Essec)
- Titre : Bayesian
Learning of
Impacts of Self-Exciting Jumps in Returns and
Volatility
- Résumé
:
The
paper proposes a new class of continuous-time asset
pricing models where
negative jumps play a
crucial role. Whenever there is a negative jump in
asset returns, it is
simultaneously passed on to diffusion variance
and the jump
intensity, generating self-exciting co-jumps of prices and
volatility and
jump clustering. To properly deal with parameter
uncertainty and
in-sample over-fitting, a Bayesian learning approach
combined with an
efficient particle filter is employed. It not only
allows for
comparison of both nested and non-nested models, but also
generates all
quantities necessary for sequential model analysis.
Empirical
investigation using S&P 500 index returns shows that
volatility jumps at the
same time as negative jumps in asset returns
mainly through jumps
in diffusion volatility. We find substantial
evidence
for jump clustering, in particular, after the recent
financial crisis in
2008, even though parameters driving dynamics of
the jump
intensity remain difficult to identify.
- Référence
: http://www.andrasfulop.com/home/Self_Exciting_Dec14.pdf?attredirects=0
- Jeudi 8
décembre
[salle 314]
- [15h] Orateur
: S.
Le Corff
(LTCI)
- Titre : Block online EM for
hidden Markov models
with application to parameter estimation in general
state-spaces
-
Résumé
: The EM algorithm has
been successfully applied for ML inference in HMM.
However, when processing
large data sets or data streams, the EM algorithm
might become
computationally very intensive and different online variants
have been recently
proposed. Despite the encouraging first results,
the convergence of
these algorithms remain an open question in the common
framework of general
state-spaces. In this contribution, we propose a new
online EM algorithm,
the block online EM algorithm. In this block online EM
algorithm, the
M-step (and thus, the update of the parameter) occurs at
some deterministic
times known in advance.The k-th (block) E-step consists
in the computation
of a mean value of the sufficient statistics associated
to the
observations in the k-th block. This algorithm
relies on the
ability to compute this
mean value sequentially, i.e. without storing all the
observations. At the
end of the block, the parameter is updated based on
a weighted average
of all the mean statistics (over the successive blocks).
The convergence
properties of this algorithm are addressed for HMM
models for which
expectations under the joint smoothing distribution
can be
computed explicitly (e.g.
finite state space or linear Gaussian model) and for a
stochastic variant
based on Sequential Monte Carlo methods in the case
of more general
models. The behavior of the proposed algorithms is
numerically
evaluated for different models.
- [16h15] Orateur
: T.
Lelièvre (CERMICS)
- Titre :
Generating efficiently metastable dynamics
- Résumé
:
We will present two methods to sample metastable
trajectories of
solutions to some stochastic differential equations
which are used in
molecular dynamics. Naive approaches are typically
inefficient since
transitions between two given metastable states are
very rare. The
interest of such sampling techniques is to obtain
typical transition
paths between two metastable regions, to compute some
reaction rates,
to determine the transition states, etc.
The first method is an adaptation of the Adaptive
Multilevel Splitting
approach proposed by F. Cérou and A.
Guyader in 2007. The
idea is to select trajectories which go the most far
in some given
direction. We have recently used this technique on
simple test cases,
and most of the mathematical analysis remains to be
done [1].
The second method (the parallel replica method) is an
algorithm which
has been proposed by A. Voter in 1997. It is in
particular used in
molecular dynamics simulations in material sciences.
We have recently
analyzed mathematically this algorithm [2]. The quasi
stationary
distribution plays a crucial role in the analysis.
[1] F. Cérou, A. Guyader, T.
Lelièvre and D.
Pommier, A multiple
replica approach
to simulate reactive trajectories, Journal of
Chemical Physics, 134,
054108, (2011).
[2] C. Le Bris, T. Lelièvre, M. Luskin and
D. Perez, A
mathematical formalization of the
parallel replica dynamics, http://hal.archives-ouvertes.fr/hal-00596161/fr/
.
- Mardi 15 Novembre
- Exceptionnellement, le
séminaire
est
remplacé
par
une
journée
scientifique qui aura lieu Ã
Telecom ParisTech. Voir site
web
- Jeudi 13 Octobre [Amphi
Hermite]
- [15h] Orateur
: B. Miasojedow (LTCI,
Paris)
- Titre : Nonasymptotic
bounds
on the estimation error of MCMC algorithms
- Résumé
:
MCMC methods are used not only to sample from
posterior distributions
but also
to estimate expectations. We derive bounds
on the mean square error of
MCMC estimators.
Our analysis is non-asymptotic. We first establish a general
result
valid for essentially all ergodic Markov chains
encountered in
Bayesian
computation and a possibly unbounded target function
f. Assuming
minorization
condition
we
cut the trajectory by regeneration technique
into iid blocks. In further analysis we use tools of
statistical sequential
analysis
and
renewal
theory, such as Wald's identities and Lorden's
theorem.
The bound is sharp in the sense that the leading term is
equal
to the well known aspymptotic results. Next,
we proceed to specific assumptions (drift and
minorization conditions)
and
give
explicit computable bounds for geometrically and polynomially
ergodic
Markov chains. In both cases final results are in terms
of
computable
quantities which appear in the drift and minorization conditions. The
talk
is based on a joint paper with K. Latuszynski and W.
Niemiro.
- [16h15] : Pas de
séminaire /
Réunion du projet BigMC
- Jeudi 1er Septembre
[salle 201]
- Orateur : X-L.
Meng (Univ. Harvard, USA)
- Titre : Statistical Inception for
the MCMC Dream:
The kick is
in the
residual (augmentation)!
- Résumé
: The development of MCMC
algorithms via
data augmentation (DA) or
equivalently auxiliary variables has some resemblance
to the theme plot
of the recent Hollywood
hit Inception. We MCMC designers all share essentially
the same ?3S?
dream, that is, to create algorithms that are simple,
stable, and speedy.
Within that grand dream, however, we have created a
rather complex web
of tools, with some of them producing very similar
algorithms but for
unclear reasons, or others that were thought to be of
different origins
but actually are layered when viewed from a suitable
distance. These
include conditional augmentation, marginal augmentation,
PX-DA,
partially non-centering parameterization, sandwiched
algorithms,
interweaving strategies, ASIS, etc. It turns out that there
is a simple
statistical insight that can unify essentially all these
methods
conceptually, and it also provides practical guidelines
for their DA
constructions. It is the simple concept of regression
residuals, which
are constructed to be orthogonal to the regression
functions. All
these methods in one form or another effectively
build a residual
augmentation. Given a DA distribution f(T,A), where T
is our targeted
variable (i.e., f(T) is our targeted distribution)
and A is the
augmented variable, there are two broad classes of
residuals
depending on whether we regress T on A or A on T. In this
talk we will
demonstrate how methods like conditional augmentation
and partially
non-centering parameterization build their residual
augmentations by
regressing A on T, whereas methods such as marginal
augmentation and
ASIS effectively use residual augmentations from
regressing T on A. For
either class, the attempted orthogonality helps to
reduce the
dependence among MCMC draws, and when the orthogonality
leads to true
independence as occurring in some special cases, we
reach the dream of
producing i.i.d. draws. (The talk is based on an
upcoming discussion
article, especially its rejoinder, Yu and Meng (2011,
JCGS) )
2010 - 2011
- Jeudi 9 Juin
[Salle 314]
- [15 h] Orateur
:H.
F.
Lopes (Univ. of Chicago, US)
- Titre : Parsimonious
Bayesian Factor Analysis when the Number
of
Factors is Unknown
-
Résumé
: We
introduce
a new and general set of identifiability conditions
for
factor models which handles the ordering problem
associated with
current common practice. In addition, the new class
of parsimonious
Bayesian factor analysis leads to a factor loading
matrix
representation which is an intuitive and easy to
implement factor
selection scheme. We argue that the structuring the
factor loadings
matrix is in concordance with recent trends in
applied factor analysis.
Our MCMC scheme for posterior inference makes
several improvements over
the existing alternatives while outlining various
strategies for
conditional posterior inference in a factor
selection scenario. Four
applications, two based on synthetic data and two
based on well known
real data, are introduced to illustrate the
applicability and
generality of the new class of parsimonious factor
models, as well as
to highlight features of the proposed sampling
schemes. (Joint
work with Sylvia
Fruhwirth-Schnatter, Univ. of Linz - Austria).
- [16h15] Orateur
: A. Eberle
(Univ.
Bonn, Germany)
- Titre : Mixing times of
Metropolis-adjusted
Langevin algorithms for
log-concave probability measures in high dimensions
-
Résumé
:
The Metropolis-adjusted Langevin algorithm (MALA) is
a
Metropolis-Hastings algorithm for approximate
sampling from continuous
distributions. We derive upper bounds for the
distance from equilibrium
after a finite number of steps for MALA with
semi-implicit Euler
proposals applied to log-concave perturbations of
Gaussian
measures. For sufficiently
``regular´´
perturbations, the estimates are
dimension-independent in a sense to be
specified.
- Jeudi 5 Mai
[Salle
314]
- [15 h] Orateur : M. Vihola
(Univ. Jyvaslyla,
Finlande)
- Titre : Stability of adaptive
random-walk
Metropolis algorithms
- Résumé
: Most
results
on adaptive MCMC in the literature are based on
assumptions
that require the adaptation process to be `stable' (in
a certain
sense). Such stability can rarely be established,
unless the process is
modified by introducing specific stabilisation
structures. This talk
outlines recent stability and ergodicity results for
adaptive MCMC
algorithms without such stabilisation. The key idea
behind the results
is that the ergodic averages can converge even if the
Markov kernels
gradually `lose' their ergodic properties.
- [16h15]
Orateur: B.
Bouzy
(Univ. Descartes, Paris)
- Titre : Monte-Carlo Tree
Search for the game of
Go
- Résumé
:
This talk describes Monte-Carlo Tree Search (MCTS) the
successful
technique now used in many complex games such as the
game of Go. In a
first part, the talk reminds the results achieved by
old go programs
until 2003, i.e. medium on the human scale. This
average result was
firstly caused by the game tree size that forbids
global tree search.
Building an accurate evaluation functions for non
terminal positions
was the second difficulty. These difficulties lead to
erratic behaviour
of go programs until 2003. In a second part, the talk
shows how the
Monte-Carlo (MC) approach has simplified the computer
go between 2003
and 2006. To evaluate a non terminal position, the MC
method consists
in launching random games starting from this position,
and playing them
out until terminal positions, easy to score, are
reached. The MC
evaluation of a position is the mean of the scores
encountered. MC
evaluations cost a lot of time, but they are robust,
and their
precision improves with the time available. In a third
part, the talk
shows the success of MCTS since 2006. MCTS cleverly
integrates tree
search and simulations. MCTS progressively builds a
tree whose nodes
contain statistics about simulations. While thinking
time is available,
MCTS repeatedly executes four stages: the selection
stage (browsing the
tree from the root to a leaf), the expansion stage
(adding a new node),
the simulation stage (performing the simulation until
the end of the
game et getting the outcome) and the update stage
(updating node
values). MCTS resulted in a big improvement of Go
programs. On 9x9
boards the best playing programs reached human
professional level. On
19x19 boards, the best playing programs reached human
strong amator
level. In a last part, the talk shows how MCTS can be
refined. First,
using domain-dependent knowledge is advised in the
selection and
simulation stages. Reinforcement learning or
supervised learning can be
used to automatically learn this knowledge. Second,
the selection stage
can be very efficient with the RAVE (Rapid Action
Value Estimation)
heuristic. Parallelization is a third mandatory tool
to demonstrate the
power of MCTS in go. Before conclusion, the talk shows
the remaining
obstacles and future works.
- Jeudi 7 Avril
[Salle 314]
- [15 h] Orateur : O.
Papaspiliopoulos
(Univ. P. Fabra, Barcelone)
- Titre :
Bayesian inference of coefficients of diffusion
processes using
data augmentation
-
Résumé
:
We consider estimation of scalar functions which
determine the dynamics
of diffusion processes. It is known that
nonparametric maximum
likelihood is ill-posed in this context. We adopt a
probabilistic
approach to regularize the problem by the adoption
of a prior
distribution for the unknown functional. Our first
result is that a
Bayesian Gaussian conjugate analysis for the drift
of one-dimensional
non-linear diffusions is feasible given
high-frequency data. This
is achieved by expressing the log-likelihood
as a quadratic
function of the drift, with sufficient statistics
given by the
so-called local time process and the end points of
the observed path.
We show how to implement such analysis using
flexible and
amenable to efficient computations Gaussian Markov
process priors. We
provide a simple estimator of the local time and a
finite element
method for the numerical derivation of the posterior
mean and
precision, as well as the posterior simulation
of the unknown
functional. We embed this technology in
partially observed
situations and adopt a data augmentation approach
whereby we
iteratively generate missing data paths and draws
from the
(conditionally) Gaussian posterior distribution of
the functional
given complete data. Our methodology is applied to
estimate the drift
of models used in molecular dynamics and financial
econometrics using
high and low frequency, real and simulated, data. We
discuss extensions
to other partially observed schemes, connections to
other types of
non/semi-parametric inference, and to estimation of
vector-valued
functions.
- Joint work
with
Pokern, Roberts and Stuart.
- [16h15] Orateur
: P. Bui Quang (ONERA Palaiseau)
- Titre :
High-dimensional importance sampling: An
analysis via
the Laplace method
- Résumé
: Several
authors
have underlined that importance sampling for Bayesian
inference
often behaves poorly when the dimension of the
parameter of interest is
large. We consider here the asymptotic variance of the
importance
weights to quantify the impact of dimensionality on
the inference. To
calculate this variance, we propose to use the Laplace
method, which
allows us to approximate accurately multidimensional
integrals. When
applied in a Bayesian framework, the approximation
converges with
respect to the data size. We use this method to
calculate the
asymptotic variance of the importance weights, and in
particular to
analyze its dependence on the dimension.
- Travail en
collaboration avec C. Musso (ONERA Palaiseau)
et F. Le Gland
(INRIA Rennes).
- Jeudi 3 Mars
[Salle 201]
- [15 h] Orateur
: O.
Feron
(EDF)
- Titre :
Echantillonnage de champs gaussiens de grande
dimension : le cas non
creux / non circulant
- Résumé
:
Ces travaux ont été
menés par
François Orieux, Post-doc Ã
l'institut
Pasteur,
Jean-François Giovannelli, professeur au
laboratoire de l'intégration
du
matériau au système de
l'université de Bordeaux, et moi-même. Dans un
premier temps, nous
proposons une nouvelle approche pour l'échantillonnage
de
champs
gaussiens
corrélés dans le cas
où les approches
classiques
ne sont pas utilisables : lorsque la dimension du problème
est
très grande et lorsque la matrice inverse
de covariance (ou
matrice de
précision) n'est pas creuse et n'est pas
circulante. Cette
approche est valide dans le
cas où une structure
particulière de la matrice de
précision est disponible. Cette structure
apparaît dans la résolution
de
problèmes inverses par des
méthodes
d'estimation bayésienne. L'algorithme
proposé trouve
une
application directe pour les méthodes
d'inversion
myopes et/ou
non
supervisées, fondées sur des
méthodes d'échantillonnage
de type MCMC. L'efficacité
de
cette approche est illustrée sur
l'inversion non
supervisée d'un
problème de super résolution
d'images. Aujourd'hui,
nous
étudions des extensions Ã
cet algorithme dans
le but d'accélérer
les
algorithmes
MCMC
mis
en
oeuvre. Dans ce but, nous énoncerons
les
différentes pistes
envisagées et les
difficultés rencontrées
pour
assurer la convergence des algorithmes MCMC.
- [16h15] Orateur : O.
Cappé
(LTCI)
- Titre
:
L'algorithme EM en
ligne et ses
possibles extensions Monte Carlo
- Résumé
:
Dans cet exposé, je
décrirai le principe de l'algorithme EM
en ligne (ou
adaptatif) qui a pour but d'estimer les
paramètres d'un
modèle à données
latentes (a
priori à partir d'observations
indépendantes)
en incorporant les observations une Ã
une. Je
présenterai également une
façon
légèrement
différente d'utiliser
l'algorithme ayant pour but de déterminer
l'estimateur du
maximum de vraisemblance correspondant Ã
un ensemble
fixé d'observations (ce qui en fait un
concurrent direct
de l'algorithme EM classique). Après une
discussion des
propriétés de convergence de
l'algorithme, je
présenterai des travaux en cours sur des
extensions de
l'approche en particulier à des cas
où
l'étape E de l'algorithme EM en ligne
nécessite l'utilisation de simulations
Monte Carlo.
- Réf. O.
Cappé and E. Moulines. On-line
Expectation-Maximization
Algorithm for Latent Data Models. J. Royal Statist.
Soc. B,
71(3):593-613, 2009.
- Jeudi 3
Février [Salle 314]
- [15h] Orateur :
F.
Perron (Univ. de Montréal, Canada)
- Titre :
Estimation de copules, une approche
bayésienne
- Résumé :
La copule associée à un
couple
aléatoire continu (X,Y) est une mesure de
dépendance qui est invariante pour les
transformations
monotones sur X et Y. Cette mesure a une forme
particulière
dans le cas des valeurs extrêmes qui fait
intervenir la
fonction de dépendance de Pickands
( fonction convex sur [0,1] avec conditions
aux bornes ). On
cherche
à estimer la fonction de Pickands. Pour
cela nous allons
constuire des
modèles bayésiens. Ces
modèles
seront motivés par une
construction
géométrique, un lien avec la
mesure spectrale
et une
utilisation des polynômes. L`aspect
computationnel va
prendre une
place
très importante. Nous allons discuter du
comment faire les
calculs en faisant appel
à des méthodes de simulation
basées sur les chaînes
de
Markov. Finalement, on traitera le
problème standard
où
les
variables ne sont pas à valeurs
extrêmes.
- [16h15] GdT : J.
Rousseau
(Univ. Dauphine / CREST)
- Référence : Y.
Pokern, O.
Papaspiliopoulos, G.O. Roberts ans A. Stuart.
Nonparametric Bayesian
drift estimation for one-dimensional diffusion
processes, submitted
to the Ann.Statist. Available
in pdf
- Jeudi 6 Janvier
[Salle 314]
- [15h] Orateur
: J.L.
Foulley
(INRA)
- Titre : Calcul
de la vraisemblance marginale par la
méthode des
posteriors de puissance : rappels et
applications
- Résumé : Cet exposé se
place dans le
cadre des méthodes de calcul de la
vraisemblance marginale
après intégration des
paramètres
par rapport à leur
densité a priori.
Après un bref rappel des principales
méthodes
standard (Importance Sampling, Newton &
Raftery, Gelfand & Dey,
Chib, « Bridge
sampling »
notamment), l’exposé se
focalisera sur la
méthode des posteriors de puissance
de
température due à Friel
& Pettitt (2008).
Nous rappellerons les fondements
théoriques de la
méthode et discuterons de sa mise en
Å“uvre
calculatoire. Nous évoquerons ses
liens avec
l’approche du facteur de Bayes
fractionnaire d’O Hagan
et avec le critère DIC. Nous
montrerons comment cette
méthode peut être
implantée sous
Winbugs. Nous illustrerons le propos par deux
exemples :
l’un relatif Ã
l’analyse de
données longitudinales par des
modèles mixtes,
et l’autre, ayant trait Ã
l’analyse de
différenciation
génétique de
marqueurs moléculaires SNP par des
modèles
bayésiens hiérarchiques
simples.
- Référence : Friel N, Pettitt AN
(2008) Marginal
likelihood estimation via power posteriors,
JRSS, B, 70, 589-607
- [16h15] Orateur
: G.
Celeux (INRIA, Futur)
- Titre :
Méthodes d'estimation pour le
modèle des blocs
latents.
- Résumé
:
Le modèle des blocs latents est un
modèle de
mélange proposé
par
Govaert et Nadif (2007) qui permet la classification conjointe
des lignes et des
colonnes d'un tableau de données. Après
sa
présentation, nous comparerons
différentes
méthodes d'estimation
des
paramètres de ce modèle pour
lequel
l'inférence pose
des
difficultés. Nous évoquerons
une
approximation variationnelle de
l'algorithme
EM (Govert et Nadif 2007), une version stochastique de EM de type
EM Ã
la Gibbs et une approche bayésienne
proposée
par Wyse et
Friel (2010).
Pour finir le problème du choix d'un
nombre de blocs
sera
évoqué et ses
particularités
souligné.
- Govaert and Nadif (2007) Block Clustering
with
Bernoulli miture models: Comparison
of
different approaches. Computational Statistics and
Data Analysis
52, 3233-3245.
- Wyse
and
Friel (2010) Block clustering with collapsed latent
block model. In
revision for Statistics
and Computing
- Jeudi 2
Décembre [Salle 314]
- Jeudi 18 Novembre
- [15h] Orateur : D.
Woodard
(Cornell Univ., USA)
- Titre : Convergence rate of
Markov chain methods
for genomic motif discovery
- Résumé
:
We analyze the convergence rate
of a popular
Gibbs sampling method used for statistical discovery
of gene regulatory
binding motifs in DNA sequences. This sampler
satisfies a very strong
form of ergodicity (uniform ergodicity). However, we
show that, due to
multimodality of the posterior distribution, the rate
of convergence
often decreases exponentially in the length of the DNA
sequence.
Specifically, we show exponential or polynomial decay
for several
cases, which support the conjecture that the decay is
exponential if
and only if more than one true repeating pattern
exists in the data.
Since there are typically multiple, even numerous,
such patterns in
real data (the goal being to detect well-conserved and
frequently-occurring patterns), we argue that the
Gibbs sampler rate of
convergence decays exponentially in practice.
This matches empirical results, which find such poor
mixing of the
motif-discovery Gibbs sampler that it is used only for
discovering
modes of the posterior distribution (candidate
motifs). This is one of
the first examples of a Markov chain method that
provably fails to
obtain samples from the posterior distribution of a
statistical model
within polynomial time.
- [16h15] GdT
:
C.
Schäfer
(CREST, Paris)
- Titre : Adaptive
Monte Carlo on
multivariate binary sampling spaces
- Résumé
: A
Monte Carlo algorithm is said to be adaptive if it can
adjust
automatically its current proposal distribution using
past simulations.
The choice of the parametric family that defines the
set of proposal
distributions is critical for good performance. In
this paper, we
discuss such parametric families for adaptive sampling
on multivariate
binary spaces. A practical motivation for this problem
is variable
selection in a linear regression context. We want to
sample from a
Bayesian posterior distribution on the model space
using Sequential
Monte Carlo for sampling and the Cross-Entropy method
for optimization.
Raw versions of the algorithm are easily implemented
using binary
vectors with independent components. For
high-dimensional variable
selection problems, however, these straightforward
proposals do not
yield satisfactory results. The key to efficient
adaptive algorithms
are binary parametric families which take correlations
into account,
analogously to the multivariate normal distribution on
continuous
spaces.
- Article : arXiv
- Jeudi 7 Octobre
[Salle 314]
- [15h]
Orateur : A. Samson
(Paris V)
- Titre :
Estimation pour des modeles mixtes
définis
par équations différentielles
stochastiques
via l'algorithme SAEM et un filtre particulaire
- Résumé
:
Les données longitudinales biologiques
peuvent
être analysées par des modèles
mixtes
définis
par
une équation différentielle
stochastique (EDS).
L'estimation par
maximum de vraisemblance dans ces modeles est complexe,
la vraisemblance
n'étant pas explicite. L'algorithme SAEM (Delyon et
al, 1999) a
été proposé
pour l'estimation
des modeles mixtes
mais
n'est pas adapté au cas des modeles
définis
par EDS. Dans ce
travail,
nous proposons de coupler l'algorithme SAEM avec un algorithme
PMCMC (Andrieu et
al, 2010), ce qui permet de realiser de facon
efficace l'etape E de
l'algorithme EM. Nous montrons la convergence
de notre algorithme vers le maximum de
vraisemblance. Nous illustrons
notre
approche sur deux exemples, un processus d'Ornstein-Ulhenbeck
et un
modele à volatilité
stochastique non
homogène en temps
très utilisé pour
modéliser des
courbes de croissance.
- [16h15] Orateur : G.
Fort (LTCI,
CNRS)
- Groupe de
Travail
: Approximation stochastique et Algorithmes MCMC
adaptatifs
- Résumé
:
Nous présenterons quelques
résultats
récents sur la convergence
d'échantillonneurs
MCMC adaptatifs lorsque le mécanisme
d'adaptation se fait
selon une dynamique d'approximation stochastique. Nous
verrons que la
convergence du processus d'adaptation n'est pas
nécessairement requise, ni
même sa
stabilité. Nous illustrerons nos propos
en
considérant l'algorithme "adaptive
Metropolis"
proposé par Haario et al. (1999); et
l'algorithme
Equi-Energy proposé par Kou et al.
(2006).
Discussion à partir des
résultats
énoncés dans
- M. Vihola (2010) On
the
Stability and Ergodicity of an Adaptive Scaling
Metropolis
Algorithm, Preprint [format
pdf]
- E. Saksman and M. Vihola (2010) On the Ergodicity of
the Adaptive
Metropolis Algorithm on Unbounded Domains ,
Ann.Appl. Probab. [format
pdf]
- G. Fort, E. Moulines et P. Priouret (2010) Convergence of
adaptive MCMC algorithms:
ergodicity and law of large numbers, Preprint
[format pdf]
