1st Edition

Mean Field Simulation for Monte Carlo Integration





ISBN 9781466504059
Published May 20, 2013 by Chapman and Hall/CRC
626 Pages 9 B/W Illustrations

USD $175.00

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Book Description

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.

Table of Contents

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

Bibliography

Index

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Author(s)

Biography

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

Reviews

"…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