11 research outputs found
Generalized Rejection Sampling Schemes and Applications in Signal Processing
Bayesian methods and their implementations by means of sophisticated Monte
Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters,
have become very popular in signal processing over the last years. However, in
many problems of practical interest these techniques demand procedures for
sampling from probability distributions with non-standard forms, hence we are
often brought back to the consideration of fundamental simulation algorithms,
such as rejection sampling (RS). Unfortunately, the use of RS techniques
demands the calculation of tight upper bounds for the ratio of the target
probability density function (pdf) over the proposal density from which
candidate samples are drawn. Except for the class of log-concave target pdf's,
for which an efficient algorithm exists, there are no general methods to
analytically determine this bound, which has to be derived from scratch for
each specific case. In this paper, we introduce new schemes for (a) obtaining
upper bounds for likelihood functions and (b) adaptively computing proposal
densities that approximate the target pdf closely. The former class of methods
provides the tools to easily sample from a posteriori probability distributions
(that appear very often in signal processing problems) by drawing candidates
from the prior distribution. However, they are even more useful when they are
exploited to derive the generalized adaptive RS (GARS) algorithm introduced in
the second part of the paper. The proposed GARS method yields a sequence of
proposal densities that converge towards the target pdf and enable a very
efficient sampling of a broad class of probability distributions, possibly with
multiple modes and non-standard forms
Space-sequential particle filters for high-dimensional dynamical systems described by stochastic differential equations
We introduce a novel methodology for particle filtering in dynamical systems
where the evolution of the signal of interest is described by a SDE and
observations are collected instantaneously at prescribed time instants. The new
approach includes the discretisation of the SDE and the design of efficient
particle filters for the resulting discrete-time state-space model. The
discretisation scheme converges with weak order 1 and it is devised to create a
sequential dependence structure along the coordinates of the discrete-time
state vector. We introduce a class of space-sequential particle filters that
exploits this structure to improve performance when the system dimension is
large. This is numerically illustrated by a set of computer simulations for a
stochastic Lorenz 96 system with additive noise. The new space-sequential
particle filters attain approximately constant estimation errors as the
dimension of the Lorenz 96 system is increased, with a computational cost that
increases polynomially, rather than exponentially, with the system dimension.
Besides the new numerical scheme and particle filters, we provide in this
paper a general framework for discrete-time filtering in continuous-time
dynamical systems described by a SDE and instantaneous observations. Provided
that the SDE is discretised using a weakly-convergent scheme, we prove that the
marginal posterior laws of the resulting discrete-time state-space model
converge to the posterior marginal posterior laws of the original
continuous-time state-space model under a suitably defined metric. This result
is general and not restricted to the numerical scheme or particle filters
specifically studied in this manuscript
Adaptive Importance Sampling: The past, the present, and the future
International audienc
Mobile Lightweight Wireless SystemsThird International ICST Conference, MOBILIGHT 2011, Bilbao, Spain, May 9-10, 2011, Revised Selected Papers /
369 p. 148 illus.online resource