Section: New Results

Mean field interacting particle systems

Participants : Ajay Jasra, Frédéric Patras, Pierre Del Moral, Arnaud Doucet, Sylvain Rubenthaler.

The design and the mathematical analysis of genetic type and branching particle interpretations of Feynman- Kac-Schroedinger type semigroups (and vice versa) has been developed by group of researchers since the beginning of the 90's. In Bayesian statistics these sampling technology is also called sequential Monte Carlo methods. For further details, we refer to the books  [33] , [37] , [26] , and references therein. This Feynman-Kac particle methodology is increasingly identified with emerging subjects of physics, biology, and engineering science. This new theory on genetic type branching and interacting particle systems has led to spectacular results in signal processing and in quantum chemistry with precise estimates of the top eigenvalues, and the ground states of Schrodinger operators. It offers a rigorous and unifying mathematical framework to analyze the convergence of a variety of heuristic-like algorithms currently used in biology, physics and engineering literature since the beginning of the 1950's. It applies to any stochastic engineering problem which can be translated in terms of functional Feynman-Kac type measures. During the last two decades the range of application of this modern approach to Feynman-Kac models has increased revealing unexpected applications in a number of scientific disciplines including in : The analysis of Dirichlet problems with boundary conditions, financial mathematics, molecular analysis, rare events and directed polymers simulation, genetic algorithms, Metropolis-Hastings type models, as well as filtering problems and hidden Markov chains.

In the period 2008-2009, these lines of research have been developed in three different directions. To develop a concrete peer to peer interaction between applied mathematics, engineering and computer sciences, we have written two review and pedagogical book chapters on stochastic particle algorithms and their applications ([12] , [17] ). Our second line of research is concerned with the foundations and the mathematical analysis of mean field particle models. In 2009, we have developed a series of new important results such as exact propagation of chaos expansions of the law of particle blocks ([7] , [21] ), sharp and non asymptotic exponential concentration inequalities [22] , and the refined stability analysis of neutral type genetic models [6] . The third line of our research is concerned with the design of new classes of stochastic particle algorithms, including particle approximate Bayesian computation [18] , backward Feynman-Kac particle models [19] , particle approximations of fixed parameters in hidden Markov chain models [20] , and particle rare event simulation of static probability laws [16] . The details of these contributions are provided below.

Review book chapters

• In [12] , we present a review on the mean field particle theory for the numerical approximation of Feynman-Kac path integrals in the context of nonlinear filtering. We show that the conditional distribution of the signal paths given a series of noisy and partial observation data is approximated by the occupation measure of a genealogical tree model associated with mean field interacting particle model.

• In [17] , we provide a pedagogical introduction to the stochastic modeling and the theoretical analysis of stochastic particle algorithms. These lectures notes were written for the Machine Learning Summer School 08. 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.

Stochastic analysis

• In [7] , [21] , We design a theoretic tree-based functional representation of a class of discrete and continuous time Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp -mean error bounds, and laws of large numbers for U-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schroedinger semigroups are also discussed.

• The article [6] , is concerned with the long time behavior of neutral genetic population models, with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We then analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.

• The article [22] is concerned with the fluctuations and the concentration properties of a general class of discrete generation and mean field particle interpretations of non linear measure valued processes. We combine an original stochastic perturbation analysis with a concentration analysis for triangular arrays of conditionally independent random sequences, which may be of independent interest. Under some additional stability properties of the limiting measure valued processes, uniform concentration properties with respect to the time parameter are also derived. The concentration inequalities presented here generalize the classical Hoeffding, Bernstein and Bennett inequalities for independent random sequences to interacting particle systems, yielding very new results for this class of models. We illustrate these results in the context of McKean Vlasov type diffusion models, McKean collision type models of gases, and of a class of Feynman-Kac distribution flows arising in stochastic engineering sciences and in molecular chemistry.

New class of particle algorithms

• The article [18] is concerned with Approximate Bayesian computation (ABC) in inference problems where the likelihood function is intractable, or expensive to calculate. To improve over Markov chain Monte Carlo (MCMC) implementations of ABC, the use of sequential Monte Carlo (SMC) methods has recently been suggested. Effective SMC algorithms that are currently available for ABC have a computational complexity that is quadratic in the number of Monte Carlo samples and require the careful choice of simulation parame- ters. In this article an adaptive SMC algorithm is proposed which admits a computational complexity that is linear in the number of samples and determines on-the-fly the simulation parameters. We demonstrate our algorithm on a toy example and a population genetics example.

•  [19] , [20] , we design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals on-the-fly as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of fixed parameter estimation in hidden Markov chain problems ([20] ), as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes ([19] ).

• The paper [16] discusses the rare event simulation for a fixed probability law. The motivation comes from problems occurring in watermarking and fingerprinting of digital contents, which is a new application of rare event simulation techniques. We provide two versions of our algorithm, and discuss the convergence properties and implementation issues. A discussion on recent related works is also provided. Finally, we give some numerical results in watermarking context. The pair of articles [25] , [24] are concerned with the the analysis of the Fisher information matrix-based nonlinear system conversion for state estimation problems.