Mean Field Particle Samplers In Statistical Learning and Rare Event Analysis

 

ABSTRACT:

In the last three decades, there has been a dramatic increase in the use of mean field particle sampling methods as a powerful tool in real-world applications of Monte Carlo simulation in computational physics, population biology, computer sciences, and statistical machine learning. The particle simulation techniques they suggest are also called resampled and diffusion Monte Carlo methods in quantum physics, genetic and evolutionary type algorithms in computer sciences, as well as Sequential Monte Carlo methods in Bayesian statistics, and particle filters in advanced signal processing. These particle methodologies are used to approximate a flow of probability measures with an increasing level of complexity. This class of probabilistic models includes conditional distributions of signals with respect to noisy and partial observations, non-absorption probabilities in Feynman–Kac–Schrödinger type models, Boltzmann–Gibbs measures, calibration and propagation of uncertainty probabilities, as well as conditional distributions of stochastic processes in critical regimes, including quasi-invariant measures and ground state computations.

This series of lectures present an introduction to the stochastic modeling and theoretical analysis of these sophisticated probabilistic models. We shall discuss the origins and mathematical foundations of these particle stochastic methods, and applications in rare event analysis, signal processing, mathematical finance and Bayesian statistical inference. We illustrate these methods through several applications: random walk confinements, particle absorption models, nonlinear filtering, stochastic optimization, combinatorial counting and directed polymer models.

Lecture 1.  AN OVERVIEW OF MEAN FIELD PARTICLE METHODS AND THEIR APPLICATIONS 
(Tuesday October 18, 09:30-11:00, CDT Lecture Room 2)
 

The first lecture offers a brief overview of mean field type particle methodologies. In the first part, we present nonlinear Markov chain models and their mean field particle interpretations. Several illustrations are discussed including sequential Monte Carlo samplers and Feynman-Kac interacting particle samplers. The second part of the lecture is concerned with a discussion on Feynman-Kac distributions on path spaces. Lastly, we briefly discuss Ensemble Kalman type filters, interacting Kalman filters as well as multiple objects particle tracking methods.

Lecture 2. FEYNMAN-KAC PARTICLE SAMPLERS
(Tuesday October 25, 09:30-11:00, CDT Lecture Room 2)
 

The second lecture is concerned with Feynman-Kac particle samplers (a.k.a. Sequential Monte Carlo methods). We present the 5 key formulae that are used to design online particle approximations of path space measures and their normalizing constants. We illustrate these probabilistic models with a dozen of examples, including filtering, Markov confinements, self-avoiding walks, spectral approximations of Schrödinger operators, importance sampling twisted distributions, Boltzmann-Gibbs measures, and parameter inference in hidden Markov chain estimation problems.

Lecture 3. PARTICLE MARKOV CHAIN MONTE CARLO METHODS
(Tuesday November 1, 09:30-11:00, CDT Lecture Room 1)
 

The third lecture is concerned with particle Markov chain Monte Carlo methods. These advanced Markov chain methods combine mean field particle methods with Markov chain Monte Carlo techniques. The first part of the lecture presents a key unbiasedness property of Feynman-Kac particle models and a natural class of extended many-body particle distributions. The second part of the lecture discuss a duality formula that allows to designing particle-Gibbs samplers based on genealogical tree-based samplers with frozen ancestral paths. We illustrate these models in the context of parameter inference in hidden Markov chain estimation problems.

Lecture 4. SOME THEORETICAL ASPECTS
(Tuesday November 8, 09:30-11:00, CDT Lecture Room 1)
 

The fourth and last of these lectures is concerned with the mathematical foundations and the theoretical aspects of the particle samplers discussed in the first lectures. We discuss some key stochastic analysis techniques including the analysis of the stability properties of nonlinear Feynman-Kac semigroups, and local linearization and perturbation analysis of mean field particle models.

 

                                              *** DUE TO ROOM CAPACITY REGISTRATION IS REQUIRED ***

                     TO REGISTER FOR THESE LECTURES PLEASE EMAIL MARTA GUZZON: m.guzzon@imperial.ac.uk