Del Moral Pierre | The University of New South Wales (original) (raw)

Papers by Del Moral Pierre

Let P(E) be the space of probability measures on a measurable space (E, E). In this paper we intr... more Let P(E) be the space of probability measures on a measurable space (E, E). In this paper we introduce a class of non-linear Markov Chain Monte Carlo (MCMC) methods for simulating from a probability measure π ∈ P(E). Non-linear Markov kernels (e.g. Del ; Del ) K : P(E) × E → P(E) can be constructed to admit π as an invariant distribution and have superior mixing properties to ordinary (linear) MCMC kernels. However, such non-linear kernels 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 demonstrate that, under some conditions, the associated self-interacting approximations exhibit a strong law of large numbers; our proof technique is via the Poisson equation and Foster-Lyapunov conditions. We investigate the performance of our approximations with some simulations, combining the methodology with population-based Markov chain Monte Carlo (e.g. Jasra et al. ). We also provide a comparison of our methods with sequential Monte Carlo samplers when applied to a continuous-time stochastic volatility model.

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 (2004)) 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.

Probability Theory and Related Fields, 2001

Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2001

The stability properties of a class of interacting measure valued processes arising in nonlinear ... more The stability properties of a class of interacting measure valued processes arising in nonlinear filtering and genetic algorithm theory is discussed.

Journal of Computational and Graphical Statistics, 2012

While statisticians are well-accustomed to performing exploratory analysis in the modeling stage ... more While statisticians are well-accustomed to performing exploratory analysis in the modeling stage of an analysis, the notion of conducting preliminary general-purpose exploratory analysis in the Monte Carlo stage (or more generally, the model-fitting stage) of an analysis is an area that we feel deserves much further attention. Toward this aim, this article proposes a general-purpose algorithm for automatic density exploration. The proposed exploration algorithm combines and expands upon components from various adaptive Markov chain Monte Carlo methods, with the Wang–Landau algorithm at its heart. Additionally, the algorithm is run on interacting parallel chains—a feature that both decreases computational cost as well as stabilizes the algorithm, improving its ability to explore the density. Performance of this new parallel adaptive Wang–Landau algorithm is studied in several applications. Through a Bayesian variable selection example, we demonstrate the convergence gains obtained with interacting chains. The ability of the algorithm’s adaptive proposal to induce mode-jumping is illustrated through a Bayesian mixture modeling application. Last, through a two-dimensional Ising model, the authors demonstrate the ability of the algorithm to overcome the high correlations encountered in spatial models. Supplemental materials are available online.

Comptes Rendus Mathematique, 2002

... a CNRS-UMR C55830, Université P. Sabatier, 31062 Toulouse, France. b Lockheed Martin, Eagan, ... more ... a CNRS-UMR C55830, Université P. Sabatier, 31062 Toulouse, France. b Lockheed Martin, Eagan, MN, USA. c IMA, University of Minnesota, 207 Church St. SE, Minneapolis, MN 55455, USA. ... To cite this article: P. Del Moral, T. Zajic, CR Acad. Sci. Paris, Ser. ...

This paper covers stochastic particle methods for the numerical solution of the nonlinear filteri... more This paper covers stochastic particle methods for the numerical solution of the nonlinear filtering equations based on the simulation of interacting particle systems. The main contribution of this paper is to prove convergence of such approximations to the optimal filter, thus yielding what seemed to be the first convergence results for such approximations of the nonlinear filtering equations. This new treatment has been influenced primarily by the development of genetic algorithms (J. H. Holland , R. Cerf ) and secondarily by the papers of H. Kunita and L. Stettner . Such interacting particle resolutions encompass genetic algorithms. Incidentally, our models provide essential insight for the analysis of genetic algorithms with a non-homogeneous fitness function with respect to time.

The stochastic ltering problem deals with the estimation of the current state of a signal process... more The stochastic ltering problem deals with the estimation of the current state of a signal process given the information supplied by an associate process, usually called the observation process. We describe a particle algorithm designed for solving numerically discrete ...

Sequential Monte Carlo (SMC) methods are a class of importance sampling and resampling techniques... more Sequential Monte Carlo (SMC) methods are a class of importance sampling and resampling techniques designed to simulate from a sequence of probability distributions. These approaches have become very popular over the last few years to solve sequential Bayesian inference problems (e.g. . However, in comparison to Markov chain Monte Carlo (MCMC), the application of SMC remains limited when, in fact, such methods are also appropriate in such contexts (e.g. Chopin ; Del Moral et al. ). In this paper, we present a simple unifying framework which allows us to extend both the SMC methodology and its range of applications. Additionally, reinterpreting SMC algorithms as an approximation of nonlinear MCMC kernels, we present alternative SMC and iterative self-interacting approximation schemes. We demonstrate the performance of the SMC methodology on static and sequential Bayesian inference problems. E γ (x) dx. If π is a high-dimensional, non-standard distribution then, to improve the exploration ability of an algorithm, it is attractive to consider an inhomogeneous sequence of P distributions to move "smoothly" from a tractable distribution π 1 = µ 1 to the target distribution π P = π. In this case

Journal of Networks, 2001

A distinctive feature of our study is that although the pair (X, Y ) might be indexed by IR+, the... more A distinctive feature of our study is that although the pair (X, Y ) might be indexed by IR+, the actual observations take place at discrete times only: this is not for mathematical convenience, but because discrete time obser-vations arise in a natural way as soon as ...

Journal of The Royal Statistical Society Series B-statistical Methodology, 2006

This paper shows how one can use Sequential Monte Carlo methods to perform what is typically done... more This paper shows how one can use Sequential Monte Carlo methods to perform what is typically done using Markov chain Monte Carlo methods. This leads to a general class of principled integration and genetic type optimization methods based on interacting particle systems.

This paper focuses on interacting particle systems methods for solving numerically a class of Fey... more This paper focuses on interacting particle systems methods for solving numerically a class of Feynman-Kac formulae arising in the study of certain parabolic differential equations, physics, biology, evolutionary computing, nonlinear filtering and elsewhere. We have tried to give an “exposé” of the mathematical theory that is useful for analyzing the convergence of such genetic-type and particle approximating models including law of large numbers, large deviations principles, fluctuations and empirical process theory as well as semigroup techniques and limit theorems for processes. In addition, we investigate the delicate and probably the most important problem of the long time behavior of such interacting measure valued processes. We will show how to link this problem with the asymptotic stability of the corresponding limiting process in order to derive useful uniform convergence results with respect to the time parameter. Several variations including branching particle models with random population size will also be presented. In the last part of this work we apply these results to continuous time and discrete time filtering problems.

Annals of Applied Probability, 2005

In this paper an original interacting particle system approach is developed for studying Markov c... more In this paper an original interacting particle system approach is developed for studying Markov chains in rare event regimes. The proposed particle system is theoretically studied through a genealogical tree interpretation of Feynman--Kac path measures. The algorithmic implementation of the particle system is presented. An estimator for the probability of occurrence of a rare event is proposed and its variance is computed, which allows to compare and to optimize different versions of the algorithm. Applications and numerical implementations are discussed. First, we apply the particle system technique to a toy model (a Gaussian random walk), which permits to illustrate the theoretical predictions. Second, we address a physically relevant problem consisting in the estimation of the outage probability due to polarization-mode dispersion in optical fibers.

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. Such systems have arisen in such diverse scienti c disciplines as physics and signal processing. We give conditions for the so-called particle density pro les 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 system arising in engineering and particularly in nonlinear ltering problems. Our second objective is to use these results to solve numerically the nonlinear ltering equation. Examples arising in uid mechanics are also given.

This article is concerned with the analysis of a new class of advanced particle Markov chain Mont... more This article is concerned with the analysis of a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by C. Andrieu, A. Doucet, and R. Holenstein. We present a natural interpretation of these models in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This new perspective sheds a new light on the foundations and the mathematical analysis of this class of models, including their propagation of chaos properties. In the process, we also present a new stochastic differential calculus based on geometric combinatorial techniques to derive explicit Taylor type expansions of the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures w.r.t. the population size of the auxiliary particle sampler. These results provide sharp quantitative estimates of the convergence properties of conditional particle Markov chain models, including sharp estimates of the contraction coefficient of conditional particle samplers, and explicit and non asymptotic L p -mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework develop in this article also allows to design new natural extensions of models including island type particle methodologies.