83 research outputs found
Particle-kernel estimation of the filter density in state-space models
Sequential Monte Carlo (SMC) methods, also known as particle filters, are
simulation-based recursive algorithms for the approximation of the a posteriori
probability measures generated by state-space dynamical models. At any given
time , a SMC method produces a set of samples over the state space of the
system of interest (often termed "particles") that is used to build a discrete
and random approximation of the posterior probability distribution of the state
variables, conditional on a sequence of available observations. One potential
application of the methodology is the estimation of the densities associated to
the sequence of a posteriori distributions. While practitioners have rather
freely applied such density approximations in the past, the issue has received
less attention from a theoretical perspective. In this paper, we address the
problem of constructing kernel-based estimates of the posterior probability
density function and its derivatives, and obtain asymptotic convergence results
for the estimation errors. In particular, we find convergence rates for the
approximation errors that hold uniformly on the state space and guarantee that
the error vanishes almost surely as the number of particles in the filter
grows. Based on this uniform convergence result, we first show how to build
continuous measures that converge almost surely (with known rate) toward the
posterior measure and then address a few applications. The latter include
maximum a posteriori estimation of the system state using the approximate
derivatives of the posterior density and the approximation of functionals of
it, for example, Shannon's entropy.
This manuscript is identical to the published paper, including a gap in the
proof of Theorem 4.2. The Theorem itself is correct. We provide an {\em
erratum} at the end of this document with a complete proof and a brief
discussion.Comment: IMPORTANT: This manuscript is identical to the published paper,
including a gap in the proof of Theorem 4.2. The Theorem itself is correct.
We provide an erratum at the end of this document. Published at
http://dx.doi.org/10.3150/13-BEJ545 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Two adaptive rejection sampling schemes for probability density functions log-convex tails
Monte Carlo methods are often necessary for the implementation of optimal
Bayesian estimators. A fundamental technique that can be used to generate
samples from virtually any target probability distribution is the so-called
rejection sampling method, which generates candidate samples from a proposal
distribution and then accepts them or not by testing the ratio of the target
and proposal densities. The class of adaptive rejection sampling (ARS)
algorithms is particularly interesting because they can achieve high acceptance
rates. However, the standard ARS method can only be used with log-concave
target densities. For this reason, many generalizations have been proposed.
In this work, we investigate two different adaptive schemes that can be used
to draw exactly from a large family of univariate probability density functions
(pdf's), not necessarily log-concave, possibly multimodal and with tails of
arbitrary concavity. These techniques are adaptive in the sense that every time
a candidate sample is rejected, the acceptance rate is improved. The two
proposed algorithms can work properly when the target pdf is multimodal, with
first and second derivatives analytically intractable, and when the tails are
log-convex in a infinite domain. Therefore, they can be applied in a number of
scenarios in which the other generalizations of the standard ARS fail. Two
illustrative numerical examples are shown
Nudging the particle filter
We investigate a new sampling scheme aimed at improving the performance of
particle filters whenever (a) there is a significant mismatch between the
assumed model dynamics and the actual system, or (b) the posterior probability
tends to concentrate in relatively small regions of the state space. The
proposed scheme pushes some particles towards specific regions where the
likelihood is expected to be high, an operation known as nudging in the
geophysics literature. We re-interpret nudging in a form applicable to any
particle filtering scheme, as it does not involve any changes in the rest of
the algorithm. Since the particles are modified, but the importance weights do
not account for this modification, the use of nudging leads to additional bias
in the resulting estimators. However, we prove analytically that nudged
particle filters can still attain asymptotic convergence with the same error
rates as conventional particle methods. Simple analysis also yields an
alternative interpretation of the nudging operation that explains its
robustness to model errors. Finally, we show numerical results that illustrate
the improvements that can be attained using the proposed scheme. In particular,
we present nonlinear tracking examples with synthetic data and a model
inference example using real-world financial data
Adapting the Number of Particles in Sequential Monte Carlo Methods through an Online Scheme for Convergence Assessment
Particle filters are broadly used to approximate posterior distributions of
hidden states in state-space models by means of sets of weighted particles.
While the convergence of the filter is guaranteed when the number of particles
tends to infinity, the quality of the approximation is usually unknown but
strongly dependent on the number of particles. In this paper, we propose a
novel method for assessing the convergence of particle filters online manner,
as well as a simple scheme for the online adaptation of the number of particles
based on the convergence assessment. The method is based on a sequential
comparison between the actual observations and their predictive probability
distributions approximated by the filter. We provide a rigorous theoretical
analysis of the proposed methodology and, as an example of its practical use,
we present simulations of a simple algorithm for the dynamic and online
adaption of the number of particles during the operation of a particle filter
on a stochastic version of the Lorenz system
A generalization of the adaptive rejection sampling algorithm
The original publication is available at www.springerlink.comRejection sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. These samples are either accepted or rejected depending on a test involving the ratio of the target and proposal densities. The adaptive rejection sampling method is an efficient algorithm to sample from a log-concave target density, that attains high acceptance rates by improving the proposal density whenever a sample is rejected. In this paper we introduce a generalized adaptive rejection sampling procedure that can be applied with a broad class of target probability distributions, possibly non-log-concave and exhibiting multiple modes. The proposed technique yields a sequence of proposal densities that converge toward the target pdf, thus achieving very high acceptance rates. We provide a simple numerical example to illustrate the basic use of the proposed technique, together with a more elaborate positioning application using real data.This work has been partially supported by
the Ministry of Science and Innovation of Spain (project MONIN,
ref. TEC-2006-13514-C02-01/TCM, project DEIPRO, ref. TEC-2009-
14504-C02-01 and program Consolider-Ingenio 2010 CSD2008-
00010 COMONSENS) and the Autonomous Community of Madrid
(project PROMULTIDIS-CM, ref. S-0505/TIC/0233).Publicad
Nested particle filters for online parameter estimation in discrete-time state-space Markov models
Documento depositado en el repositorio arXiv.org. VersiĂłn: arXiv:1308.1883v5 [stat.CO]We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent "sequential Monte Carlo square" (SMC2) algorithm. However, unlike the SMC2 scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in Lp (pâ„1) with convergence rate proportional to 1Nâ+1Mâ, where N is the number of Monte Carlo samples in the parameter space and NĂM is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC2 algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.The work of the D. Crisan has been partially supported by the EPSRC
grant no
EP/N023781/1. The work of J. MĂguez was partially supported by t
he Office of
Naval Research Global (award no. N62909- 15-1-2011),
Ministerio de EconomĂa y
Competitividad
of Spain (project TEC2015-69868-C2-1-R ADVENTURE) and
Ministerio
de EducaciĂłn, Cultura y Deporte
of Spain (Programa Nacional de Movilidad de Recursos
Humanos
PRX12/00690).
Part of this work was carried out while J. M. was a visitor at the Depar
tment
of Mathematics of Imperial College London, with partial support fr
om an EPSRC
Mathematics Platform grant. D. C. and J. M. would also like to acknow
ledge the support
of the Isaac Newton Institute through the program âMonte Carlo
Inference for High-Dimensional Statistical Modelsâ, as well as the constructive comme
nts of an anonymous
Reviewer, who helped improving the final version of this manuscrip
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