124 research outputs found
On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering
We analyse the exponential stability properties of a class of measure-valued
equations arising in nonlinear multi-target filtering problems. We also prove
the uniform convergence properties w.r.t. the time parameter of a rather
general class of stochastic filtering algorithms, including sequential Monte
Carlo type models and mean eld particle interpretation models. We illustrate
these results in the context of the Bernoulli and the Probability Hypothesis
Density filter, yielding what seems to be the first results of this kind in
this subject
A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces
Continuous time Feynman-Kac measures on path spaces are central in applied
probability, partial differential equation theory, as well as in quantum
physics. This article presents a new duality formula between normalized
Feynman-Kac distribution and their mean field particle interpretations. Among
others, this formula allows us to design a reversible particle Gibbs-Glauber
sampler for continuous time Feynman-Kac integration on path spaces. This result
extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein [2]
in the context of discrete generation models to continuous time Feynman-Kac
models and their interacting jump particle interpretations. We also provide new
propagation of chaos estimates for continuous time genealogical tree based
particle models with respect to the time horizon and the size of the systems.
These results allow to obtain sharp quantitative estimates of the convergence
rate to equilibrium of particle Gibbs-Glauber samplers. To the best of our
knowledge these results are the first of this kind for continuous time
Feynman-Kac measures
On the concentration properties of Interacting particle processes
These lecture notes present some new concentration inequalities for
Feynman-Kac particle processes. We analyze different types of stochastic
particle models, including particle profile occupation measures, genealogical
tree based evolution models, particle free energies, as well as backward Markov
chain particle models. We illustrate these results with a series of topics
related to computational physics and biology, stochastic optimization, signal
processing and bayesian statistics, and many other probabilistic machine
learning algorithms. Special emphasis is given to the stochastic modeling and
the quantitative performance analysis of a series of advanced Monte Carlo
methods, including particle filters, genetic type island models, Markov bridge
models, interacting particle Markov chain Monte Carlo methodologies
Numerical methods for sensitivity analysis of Feynman-Kac models
The aim of this work is to provide efficient numerical methods to estimate the gradient of a Feynman-Kac flow with respect to a parameter of the model. The underlying idea is to view a Feynman-Kac flow as an expectation of a product of potential functions along a canonical Markov chain, and to use usual techniques of gradient estimation in Markov chains. Combining this idea with the use of interacting particle methods enables us to obtain two new algorithms that provide tight estimations of the sensitivity of a Feynman-Kac flow. Each algorithm has a linear computational complexity in the number of particles and is demonstrated to be asymptotically consistent. We also carefully analyze the differences between these new algorithms and existing ones. We provide numerical experiments to assess the practical efficiency of the proposed methods and explain how to use them to solve a parameter estimation problem in Hidden Markov Models. To conclude we can say that these algorithms outperform the existing ones in terms of trade-off between computational complexity and estimation quality
Quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles
We propose a new solvable class of multidimensional quantum harmonic oscillators for a linear diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force. Under natural and easily checked controllability conditions, the ground state and the zero-point energy are explicitly computed in terms of a positive fixed point of a continuous time algebraic Riccati matrix equation. We also present an explicit solution of normalized and time dependent Feynman-Kac measures in terms of a time varying linear dynamical system coupled with a differential Riccati matrix equation. A refined non asymptotic analysis of the stability of these models is developed based on a recently developed Floquet-type representation of time varying exponential semigroups of Riccati matrices. We provide explicit and non asymptotic estimates of the exponential decays to equilibrium of Feynman-Kac semigroups in terms of Wasserstein distances or Boltzmann-relative entropy. For reversible models we develop a series of functional inequalities including de Bruijn identity, Fisher's information decays, log-Sobolev inequalities, and entropy contraction estimates. In this context, we also provide a complete and explicit description of all the spectrum and the excited states of the Hamiltonian, yielding what seems to be the first result of this type for this class of models. We illustrate these formulae with the traditional harmonic oscillator associated with real time Brownian particles and Mehler's formula. The analysis developed in this article can also be extended to solve time dependent Schrodinger equations equipped with time varying linear diffusions and quadratic potential functions
On the Stability of Positive Semigroups
International audienceThe stability and contraction properties of positive integral semigroups on Polish spaces are investigated. Our novel analysis is based on the extension of V-norm contraction methods, associated to functionally weighted Banach spaces for Markov semigroups, to positive semigroups. This methodology is applied to a general class of positive andpossibly time-inhomogeneous bounded integral semigroups and their normalised versions. The spectral theorems that we develop are an extension of Perron-Frobenius and Krein-Rutman theorems for positive operators to a class of time-varying positive semigroups. In the context of time-homogeneous models, the regularity conditions discussed in the present article appear to be necessary and sufficient condition for the existence of leading eigenvalues. We review and illustrate the impact of these results in the context of positive semigroups arising in transport theory, physics, mathematical biology and signal processing
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
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