13,390 research outputs found
Importance Sampling: Intrinsic Dimension and Computational Cost
The basic idea of importance sampling is to use independent samples from a
proposal measure in order to approximate expectations with respect to a target
measure. It is key to understand how many samples are required in order to
guarantee accurate approximations. Intuitively, some notion of distance between
the target and the proposal should determine the computational cost of the
method. A major challenge is to quantify this distance in terms of parameters
or statistics that are pertinent for the practitioner. The subject has
attracted substantial interest from within a variety of communities. The
objective of this paper is to overview and unify the resulting literature by
creating an overarching framework. A general theory is presented, with a focus
on the use of importance sampling in Bayesian inverse problems and filtering.Comment: Statistical Scienc
Uniform Time Average Consistency of Monte Carlo Particle Filters
We prove that bootstrap type Monte Carlo particle filters approximate the
optimal nonlinear filter in a time average sense uniformly with respect to the
time horizon when the signal is ergodic and the particle system satisfies a
tightness property. The latter is satisfied without further assumptions when
the signal state space is compact, as well as in the noncompact setting when
the signal is geometrically ergodic and the observations satisfy additional
regularity assumptions.Comment: 21 pages, 1 figur
Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
We examine some differential geometric approaches to finding approximate
solutions to the continuous time nonlinear filtering problem. Our primary focus
is a new projection method for the optimal filter infinite dimensional
Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric
and on a family of normal mixtures. We compare this method to earlier
projection methods based on the Hellinger distance/Fisher metric and
exponential families, and we compare the L2 mixture projection filter with a
particle method with the same number of parameters, using the Levy metric. We
prove that for a simple choice of the mixture manifold the L2 mixture
projection filter coincides with a Galerkin method, whereas for more general
mixture manifolds the equivalence does not hold and the L2 mixture filter is
more general. We study particular systems that may illustrate the advantages of
this new filter over other algorithms when comparing outputs with the optimal
filter. We finally consider a specific software design that is suited for a
numerically efficient implementation of this filter and provide numerical
examples.Comment: Updated and expanded version published in the Journal reference
below. Preprint updates: January 2016 (v3) added projection of Zakai Equation
and difference with projection of Kushner-Stratonovich (section 4.1). August
2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1)
versio
Inverse Problems and Data Assimilation
These notes are designed with the aim of providing a clear and concise
introduction to the subjects of Inverse Problems and Data Assimilation, and
their inter-relations, together with citations to some relevant literature in
this area. The first half of the notes is dedicated to studying the Bayesian
framework for inverse problems. Techniques such as importance sampling and
Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the
desirable property that in the limit of an infinite number of samples they
reproduce the full posterior distribution. Since it is often computationally
intensive to implement these methods, especially in high dimensional problems,
approximate techniques such as approximating the posterior by a Dirac or a
Gaussian distribution are discussed. The second half of the notes cover data
assimilation. This refers to a particular class of inverse problems in which
the unknown parameter is the initial condition of a dynamical system, and in
the stochastic dynamics case the subsequent states of the system, and the data
comprises partial and noisy observations of that (possibly stochastic)
dynamical system. We will also demonstrate that methods developed in data
assimilation may be employed to study generic inverse problems, by introducing
an artificial time to generate a sequence of probability measures interpolating
from the prior to the posterior
Rigorous free fermion entanglement renormalization from wavelet theory
We construct entanglement renormalization schemes which provably approximate
the ground states of non-interacting fermion nearest-neighbor hopping
Hamiltonians on the one-dimensional discrete line and the two-dimensional
square lattice. These schemes give hierarchical quantum circuits which build up
the states from unentangled degrees of freedom. The circuits are based on pairs
of discrete wavelet transforms which are approximately related by a
"half-shift": translation by half a unit cell. The presence of the Fermi
surface in the two-dimensional model requires a special kind of circuit
architecture to properly capture the entanglement in the ground state. We show
how the error in the approximation can be controlled without ever performing a
variational optimization.Comment: 15 pages, 10 figures, one theore
On Quantum Statistical Inference, I
Recent developments in the mathematical foundations of quantum mechanics have
brought the theory closer to that of classical probability and statistics. On
the other hand, the unique character of quantum physics sets many of the
questions addressed apart from those met classically in stochastics.
Furthermore, concurrent advances in experimental techniques and in the theory
of quantum computation have led to a strong interest in questions of quantum
information, in particular in the sense of the amount of information about
unknown parameters in given observational data or accessible through various
possible types of measurements. This scenery is outlined (with an audience of
statisticians and probabilists in mind).Comment: A shorter version containing some different material will appear
(2003), with discussion, in J. Roy. Statist. Soc. B, and is archived as
quant-ph/030719
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