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Approximate Bayesian computation for smoothing
 Stoch. Anal. Appl
, 2014
"... We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of au ..."
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We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of auxiliary variables valued in the same space as the observations. The quality of the approximation may be controlled to arbitrary precision through a parameter > 0. We provide theoretical results which quantify, in terms of , the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions. Under regularity assumptions, this error is O(n), where n is the number of time steps over which smoothing is performed. For numerical implementation we adopt the forwardonly sequential Monte Carlo (SMC) scheme of [16] and quantify the combined error from the ABC and SMC approximations. This forms some of the first quantitative results for ABC methods which jointly treat the ABC and simulation errors, with a finite number of data and simulated samples. When the HMM has unknown static parameters, we consider particle Markov chain Monte Carlo [2] (PMCMC) methods for batch statistical inference.
AN ITERATIVE IMPLEMENTATION OF THE IMPLICIT NONLINEAR FILTER
"... Abstract. Implicit sampling is a sampling scheme for particle filters, designed to move particles onebyone so that they remain in highprobability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations. 1991 Mathem ..."
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Abstract. Implicit sampling is a sampling scheme for particle filters, designed to move particles onebyone so that they remain in highprobability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations. 1991 Mathematics Subject Classification. 60G35, 62M20, 86A05. The dates will be set by the publisher. 1.
INTERPOLATION AND ITERATION FOR NONLINEAR FILTERS
"... Abstract. We present a general form of the iteration and interpolation process used in implicit particle filters. Implicit filters are based on a pseudoGaussian representation of posterior densities, and are designed to focus the particle paths so as to reduce the number of particles needed in nonl ..."
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Abstract. We present a general form of the iteration and interpolation process used in implicit particle filters. Implicit filters are based on a pseudoGaussian representation of posterior densities, and are designed to focus the particle paths so as to reduce the number of particles needed in nonlinear data assimilation. Examples are given. 1991 Mathematics Subject Classification. 60G35, 62M20, 86A05. The dates will be set by the publisher. 1.
NonBayesian particle filters
, 905
"... Particle filters for data assimilation in nonlinear problems use “particles” (replicas of the underlying system) to generate a sequence of probability density functions (pdfs) through a Bayesian process. This can be expensive because a significant number of particles has to be used to maintain accur ..."
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Particle filters for data assimilation in nonlinear problems use “particles” (replicas of the underlying system) to generate a sequence of probability density functions (pdfs) through a Bayesian process. This can be expensive because a significant number of particles has to be used to maintain accuracy. We offer here an alternative, in which the relevant pdfs are sampled directly by an iteration. An example is discussed in detail. Keywords particle filter, chainless sampling, normalization factor, iteration, nonBayesian 1 Introduction. There are many problems in science in which the state of a system must be identified from an uncertain equation supplemented by a stream of noisy data (see e.g. [1]). A natural model of this situation consists of a stochastic differential equation (SDE):
DOI: 10.1051/m2an/2011055 www.esaimm2an.org AN ITERATIVE IMPLEMENTATION OF THE IMPLICIT NONLINEAR FILTER
, 2009
"... Abstract. Implicit sampling is a sampling scheme for particle filters, designed to move particles onebyone so that they remain in highprobability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations. ..."
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Abstract. Implicit sampling is a sampling scheme for particle filters, designed to move particles onebyone so that they remain in highprobability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations.