Papers by Pierre Del Moral
arXiv (Cornell University), 2010
Sequential Monte Carlo (SMC) methods are a widely used set of computational tools for inference i... more Sequential Monte Carlo (SMC) methods are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. We propose a new SMC algorithm to compute the expectation of additive functionals recursively. Essentially, it is an online or forward-only implementation of a forward filtering backward smoothing SMC algorithm proposed in Doucet .et .al (2000). Compared to the standard path space SMC estimator whose asymptotic variance increases quadratically with time even under favourable mixing assumptions, the asymptotic variance of the proposed SMC estimator only increases linearly with time. This forward smoothing procedure allows us to implement on-line maximum likelihood parameter estimation algorithms which do not suffer from the particle path degeneracy problem.
The Annals of Applied Probability, 1998
In the paper we study interacting particle approximations of discrete time and measure-valued dyn... more In the paper we study interacting particle approximations of discrete time and measure-valued dynamical systems. These systems have arisen in such diverse scientific disciplines as physics and signal processing. We give conditions for the so-called particle density profiles to converge to the desired distribution when the number of particles is growing. The strength of our approach is that is applicable to a large class of measure-valued dynamical systems arising in engineering and particularly in nonlinear filtering problems. Our second objective is to use these results to solve numerically the nonlinear filtering equation. Examples arising in fluid mechanics are also given.
Stochastic Processes and their Applications, 2017
This article provides a new theory for the analysis of the particle Gibbs (PG) sampler . Followin... more This article provides a new theory for the analysis of the particle Gibbs (PG) sampler . Following the work of Del Moral and Jasra (2017) we provide some analysis of the particle Gibbs sampler, giving first order expansions of the kernel and minorization estimates. In addition, first order propagation of chaos estimates are derived for empirical measures of the dual particle model with a frozen path, also known as the conditional sequential Monte Carlo (SMC) update of the PG sampler. Backward and forward PG samplers are discussed, including a first comparison of the contraction estimates obtained by first order estimates. We illustrate our results with an example of fixed parameter estimation arising in hidden Markov models. c
Bernoulli, 2018
We consider an elliptic and time-inhomogeneous diffusion process with timeperiodic coefficients e... more We consider an elliptic and time-inhomogeneous diffusion process with timeperiodic coefficients evolving in a bounded domain of R d with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion evolution operator; when a particle is absorbed, another selected particle splits into two offsprings. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.
Maslov optimisation theory: Stochastic interpretation, particle resolution
Lecture Notes in Control and Information Sciences, 1994
Without Abstract
Lecture Notes in Control and Information Science
This article focuses on branching particle interpretations of rare events. We connect importance ... more This article focuses on branching particle interpretations of rare events. We connect importance sampling techniques with interacting particle algorithms, and multi-splitting branching models. These Monte Carlo methods are illustrated with a variety of examples arising in particle trapping analysis, as well as in ruin type estimation problems. We also provide a rather detailed presentation of the asymptotic theory of these particle algorithms, including exponential extinction probabilities, Lp-mean error bounds, central limit theorem, and fluctuation variance comparaisons.
Modèles de fractal dans la nature
Mathématiques et Applications, 2014
Les formes fractales sont des modeles mathematiques hautement symetriques. On a beau les tourner ... more Les formes fractales sont des modeles mathematiques hautement symetriques. On a beau les tourner et les retourner dans tous les sens suivant certains angles, on retrouve toujours la meme forme. Plus fort encore, si on les observe avec une loupe aussi puissante que l’on veut, on retrouve a nouveau les memes images!
Mesures de Feynman-Kac et Méthodes Particulaires
Mathématiques et Applications, 2014
La simulation de mesure de probabilites complexes sur des espaces de grandes dimensions est l’un ... more La simulation de mesure de probabilites complexes sur des espaces de grandes dimensions est l’un des problemes majeurs de l’ingenierie stochastique.
Revista de Matemática: Teoría y Aplicaciones, 2009
We present in this work a natural Interacting Particle System (IPS) approach for modeling and stu... more We present in this work a natural Interacting Particle System (IPS) approach for modeling and studying the asymptotic behavior of Genetic Algorithms (GAs). In this model, a population is seen as a distribution (or measure) on the search space, and the Genetic Algorithm as a measure valued dynamical system. This model allows one to apply recent convergence results from the IPS literature for studying the convergence of genetic algorithms when the size of the population tends to infinity. We first review a number of approaches to Genetic Algorithms modeling and related convergence results. We then describe a general and abstract discrete time Interacting Particle System model for GAs, and we propose a brief review of some recent asymptotic results about the convergence of the N -IPS approximating model (of finite N -sized-population GAs) towards the IPS model (of infinite population GAs), including law of large number theorems, ILp-mean and exponential bounds as well as large deviatio...
Biips is a software platform for automatic Bayesian inference with interacting particle systems. ... more Biips is a software platform for automatic Bayesian inference with interacting particle systems. Biips allows users to define their statistical model in the probabilistic programming BUGS language, as well as to add custom functions or samplers within this language. Then it runs sequential Monte Carlo based algorithms (particle filters, particle independent Metropolis-Hastings, particle marginal Metropolis-Hastings) in a black-box manner so that to approximate the posterior distribution of interest as well as the marginal likelihood. The software is developed in C++ with interfaces with the softwares R, MATLAB and Octave.
ESAIM: Proceedings, 2007
In this paper we introduce a class of non-linear Markov Chain Monte Carlo (MCMC) methods for simu... more In this paper we introduce a class of non-linear Markov Chain Monte Carlo (MCMC) methods for simulating from a probability measure π. Non-linear Markov kernels (e.g. Del Moral ( )) can be constructed to admit π as an invariant distribution and have typically superior mixing properties to ordinary (linear) MCMC kernels. However, such non-linear kernels often cannot be simulated exactly, so, in the spirit of particle approximations of Feynman-Kac formulae (Del Moral 2004), we construct approximations of the non-linear kernels via Self-Interacting Markov Chains (Del Moral & Miclo 2004) (SIMC). We present several non-linear kernels and investigate the performance of our approximations with some simulations.
In this article, a genetic-type algorithm based on interacting particle systems is presented, tog... more In this article, a genetic-type algorithm based on interacting particle systems is presented, together with a genealogical model, for estimating a class of rare events arising for instance in telecommunication networks, nuclear engineering, etc. The distribution of a Markov process hitting a rare but critical set is represented in terms of a Feynman-Kac model in path space. Approximation results obtained previously for these models are applied here to estimate the probability of the rare events as well as the probability distribution of the critical trajectories.
Large deviations for interacting processes in the strong topology
Strong large deviations principles for a general class of discrete generation and interacting par... more Strong large deviations principles for a general class of discrete generation and interacting particle systems are developed. The analysis is essentially conducted through an original projective interpretation of the τ-topology, combined with a powerful projective transfer result presented by the first author with J. Gärtner. These large deviations principles simplify and encompass the ones obtained in an earlier joint work of the second author with A. Guionnet. They are illustrated with simplified versions of McKean-Vlasov diffusions, and Boltzmann type collision models. We also describe the impact of this analysis on a recently developed class of genealogical and interacting particle interpretations of non linear Feynman-Kac-Schrödinger path measures.
The aim of this paper is to present efficient algorithms for the detection of multiple targets in... more The aim of this paper is to present efficient algorithms for the detection of multiple targets in noisy images. The algorithms are based on the optimal filter of a multidimensional Markov chain signal. We also present some simulations, in the case of one, two and three targets, showing the efficiency of the method for detecting the positions of the targets.
Mathématiques et Applications, 2014
Tous droits de traduction, de reproduction et d'adaptation réservés pour tous pays. La loi du 11 ... more Tous droits de traduction, de reproduction et d'adaptation réservés pour tous pays. La loi du 11 mars 1957 interdit les copies ou les reproductions destinées à une utilisation collective. Toute représentation, reproduction intégrale ou partielle faite par quelque procédé que ce soit, sans leconsentement de l'auteur ou de ses ayants cause, est illicite et constitue une contrefaçon sanctionnée parles articles 425 et suivants du Code pénal. Springer est membre du groupe Springer Science+Business Media (www.springer.com
Application of Cognitive Techniques to Network Management and Control
Advances in Intelligent Systems and Computing, 2014
This paper describes the latest communications technologies emphasizing the need of dynamic netwo... more This paper describes the latest communications technologies emphasizing the need of dynamic network control and real-time management operations. It is advocated that many such operations can profit from cognitive learning based techniques that could drive many management or control operations. In that context a short overview of selected networking approaches like 3GPP Self Organizing Networks, Autonomic Network Management and Software-Defined Networking, with some references to existing cognitive approaches is given.
ESAIM: Proceedings, 2014
Interacting particle methods are increasingly used to sample from complex high-dimensional distri... more Interacting particle methods are increasingly used to sample from complex high-dimensional distributions. They have found a wide range of applications in applied probability, Bayesian statistics and information engineering. Understanding rigorously these new Monte Carlo simulation tools leads to fascinating mathematics related to Feynman-Kac path integral theory and their interacting particle interpretations. In these lecture notes, we provide a pedagogical introduction to the stochastic modeling and the theoretical analysis of these particle algorithms. We also illustrate these methods through several applications including random walk confinements, particle absorption models, nonlinear filtering, stochastic optimization, combinatorial counting and directed polymer models. Les méthodes particulaires en interaction sont de plus en plus utilisées pour simuler des mesures de probabilités complexes dans des espaces de grandes dimensions. Leurs domaines d'applications sont diverses et variés en probabilités appliquées, en statistique bayesienne et dans les sciences de l'ingénieur. L'analyse rigoureuse de ces nouvelles techniques de simulation de type Monte Carlo conduit à des techniques mathématiques fascinantes liées à la théorie des intégrales de Feynman et leurs interprétations particulaires. Nous présentons dans ces notes une introduction pédagogique à la modélisation stochastique et l'analyse théorique de ces algorithmes particulaires. Nous illustrons ces modèles avec différentes applications, telles le confinement de marches aléatoires, des modèles d'évolutions de particules dans des milieux absorbants, des modèles de filtrage non linéaire, des problèmes d'optimisation stochastique, des questions de comptage combinatoire et des modèles de polymères dirigés.
Stochastic Processes and their Applications, 2011
Several particle algorithms admit a Feynman-Kac representation such that the potential function m... more Several particle algorithms admit a Feynman-Kac representation such that the potential function may be expressed as a recursive function which depends on the complete state trajectory. An important example is the mixture Kalman filter, but other models and algorithms of practical interest fall in this category. We study the asymptotic stability of such particle algorithms as time goes to infinity. As a corollary, practical conditions for the stability of the mixture Kalman filter, and a mixture GARCH filter, are derived. Finally, we show that our results can also lead to weaker conditions for the stability of standard particle algorithms, such that the potential function depends on the last state only.
Stochastic Processes and their Applications, 2000
We present a weighted sampling Moran particle system model for the numerical solving of a class o... more We present a weighted sampling Moran particle system model for the numerical solving of a class of Feynman-Kac formulae which arise in di erent ÿelds. Our major motivation was from nonlinear ÿltering, but our approach is context free. We will show that under certain regularity conditions the resulting interacting particle scheme converges to the considered nonlinear equations. In the setting of nonlinear ÿltering, the L 1 -convergence exponent resulting from our proof also improves recent results on other particle interpretations of these equations.
Statistics and Computing, 2013
The approximation of the Feynman-Kac semigroups by systems of interacting particles is a very act... more The approximation of the Feynman-Kac semigroups by systems of interacting particles is a very active research field, with applications in many different areas. In this paper, we study the parallelization of such approximations. The total population of particles is divided into sub-populations, referred to as islands. The particles within each island follow the usual selection / mutation dynamics. We
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Papers by Pierre Del Moral