1st Edition

Mean Field Simulation for Monte Carlo Integration

By Pierre Del Moral Copyright 2013
    626 Pages 9 B/W Illustrations
    by Chapman & Hall

    626 Pages 9 B/W Illustrations
    by Chapman & Hall

    In the last three decades, there has been a dramatic increase in the use of interacting particle methods as a powerful tool in real-world applications of Monte Carlo simulation in computational physics, population biology, computer sciences, and statistical machine learning. Ideally suited to parallel and distributed computation, these advanced particle algorithms include nonlinear interacting jump diffusions; quantum, diffusion, and resampled Monte Carlo methods; Feynman-Kac particle models; genetic and evolutionary algorithms; sequential Monte Carlo methods; adaptive and interacting Markov chain Monte Carlo models; bootstrapping methods; ensemble Kalman filters; and interacting particle filters.

    Mean Field Simulation for Monte Carlo Integration presents the first comprehensive and modern mathematical treatment of mean field particle simulation models and interdisciplinary research topics, including interacting jumps and McKean-Vlasov processes, sequential Monte Carlo methodologies, genetic particle algorithms, genealogical tree-based algorithms, and quantum and diffusion Monte Carlo methods.

    Along with covering refined convergence analysis on nonlinear Markov chain models, the author discusses applications related to parameter estimation in hidden Markov chain models, stochastic optimization, nonlinear filtering and multiple target tracking, stochastic optimization, calibration and uncertainty propagations in numerical codes, rare event simulation, financial mathematics, and free energy and quasi-invariant measures arising in computational physics and population biology.

    This book shows how mean field particle simulation has revolutionized the field of Monte Carlo integration and stochastic algorithms. It will help theoretical probability researchers, applied statisticians, biologists, statistical physicists, and computer scientists work better across their own disciplinary boundaries.

    Monte Carlo and Mean Field Models
    Linear evolution equations
    Nonlinear McKean evolutions
    Time discretization schemes
    Illustrative examples
    Mean field particle methods

    Theory and Applications
    A stochastic perturbation analysis
    Feynman-Kac particle models
    Extended Feynman-Kac models
    Nonlinear intensity measure equations
    Statistical machine learning models
    Risk analysis and rare event simulation

    Feynman-Kac Models
    Discrete Time Feynman-Kac Models
    A brief treatise on evolution operators
    Feynman-Kac models
    Some illustrations
    Historical processes
    Feynman-Kac sensitivity measures

    Four Equivalent Particle Interpretations
    Spatial branching models
    Sequential Monte Carlo methodology
    Interacting Markov chain Monte Carlo algorithms
    Mean field interacting particle models

    Continuous Time Feynman-Kac Models
    Some operator aspects of Markov processes
    Feynman-Kac models
    Continuous time McKean models
    Mean field particle models

    Nonlinear Evolutions of Intensity Measures
    Intensity of spatial branching processes
    Nonlinear equations of positive measures
    Multiple-object nonlinear filtering equations
    Association tree-based measures

    Application Domains
    Particle Absorption Models
    Particle motions in absorbing medium
    Mean field particle models
    Some illustrations
    Absorption models in random environments

    Particle Feynman-Kac models
    Doob h-processes

    Signal Processing and Control Systems
    Nonlinear filtering problems
    Linear Gaussian models
    Interacting Kalman filters
    Quenched and annealed filtering models
    Particle quenched and annealed models
    Parameter estimation in hidden Markov models
    Optimal stopping problems

    Theoretical Aspects
    Mean Field Feynman-Kac Models

    Feynman-Kac models
    McKean-Markov chain models
    Perfect sampling models
    Interacting particle systems
    Some convergence estimates
    Continuous time models

    A General Class of Mean Field Models
    Description of the models
    Some weak regularity properties
    Some illustrative examples
    A stochastic coupling technique
    Fluctuation analysis

    Empirical Processes
    Description of the models
    Nonasymptotic theorems
    A reminder on Orlicz’s norms
    Finite marginal inequalities
    Maximal inequalities
    Cramér-Chernov inequalities
    Perturbation analysis
    Interacting processes

    Feynman-Kac Semigroups
    Description of the models
    Stability properties
    Semigroups of nonlinear Markov chain models
    Backward Markovian semigroups

    Intensity Measure Semigroups
    Spatial branching models
    Measure-valued nonlinear equations
    Weak Lipschitz properties of semigroups
    Stability properties of PHD models

    Particle Density Profiles
    Stochastic perturbation analysis
    First order expansions
    Some nonasymptotic theorems
    Fluctuation analysis
    Concentration inequalities
    A general class of mean field particle models
    Particle branching intensity measures
    Positive measure particle equations

    Genealogical Tree Models
    Some equivalence principles
    Some nonasymptotic theorems
    Ancestral tree occupation measures
    Central limit theorems
    Concentration inequalities

    Particle Normalizing Constants
    Unnormalized particle measures
    Some key decompositions
    Fluctuation theorems
    A nonasymptotic variance theorem
    Lp-mean error estimates
    Concentration analysis

    Backward Particle Markov Models
    Description of the models
    Conditioning principles
    Integral transport properties
    Additive functional models
    A stochastic perturbation analysis
    Orlicz norm and Lm-mean error estimates
    Some nonasymptotic variance estimates
    Fluctuation analysis
    Concentration inequalities




    Pierre Del Moral is a professor in the School of Mathematics and Statistics at the University of New South Wales in Sydney, Australia.

    "…I found this to be an enjoyable read. Many illustrative examples reveal intriguing paradoxes in statistical theories, some of them are well-known and complemented with a broad informative discussion and others are less obvious."
    —Journal of the American Statistical Association