Stability properties of some particle filters
type: Item
instance of: scholarly article
title: Stability properties of some particle filters
zbMATH Open document ID: 1296.60098
DOI: 10.1214/12-AAP909
author: Nick Whiteley
published in: The Annals of Applied Probability
publication date: 2014-01-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.6779
reviewed by: M. P. Moklyachuk
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 93D20
Mathematics Subject Classification ID: 93E10
Mathematics Subject Classification ID: 93E11
Mathematics Subject Classification ID: 65C40
zbMATH DE Number: 6247418
zbMATH Keywords: sequential Monte Carlo
zbMATH Keywords: filtering
zbMATH Keywords: particle filter
zbMATH Keywords: approximation
zbMATH Keywords: stability
zbMATH Keywords: Markov chain
zbMATH Keywords: hidden Markov model
MaRDI profile type: Publication
arXiv ID: 1109.6779
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OpenAlex ID: W3101274747
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review text: The author deals with filtering problems for hidden Markov models. He considers a hidden Markov model which is a bivariate, discrete-time Markov chain \(((X_n, Y_n);n\geq0)\), where the signal process \((X_n)\) is also a Markov chain, with a (noncompact) state-space \(X\), and each observation \(Y_n\), with values in the observation space \(Y\), is conditionally independent of the rest of the bi-variate process given \(X_n\). The spaces \(X\) and \(Y\) are assumed to be Polish spaces endowed with their respective Borel \(\sigma\)-algebras. For observations \((y_0,y_1,\dots)\), a recursive one-step-ahead prediction filter is defined as the sequence \(((\pi_n);n\geq0)\) of conditional distributions of \(X_n\) given \((Y_0,Y_1,\dots,Y_{n-1})=(y_0,y_1,\dots,y_{n-1})\) and \(((Z_n);n\geq0)\) -- the joint density of \((Y_0,Y_1,\dots,Y_{n-1})\) evaluated at \((y_0,y_1,\dots,y_{n-1})\). Hidden Markov models are simple and yet flexible models which have found countless applications. However, in many practical situations, \(((\pi_n);n\geq0)\) and \(((Z_n);n\geq0)\) are not available in closed form. Particle filters are a class of stochastic algorithms which yield approximations \((\pi_n^N)\) and \((Z_n^N)\) of \((\pi_n)\) and \((Z_n)\) using \(N\) samples. A large number of variations and extensions of this algorithm have been developed. However, there are still very few results which establish stability over time of particle filtering methods.NEWLINENEWLINEThe author of the present paper proves some stability properties of a standard particle filter under assumptions which are verifiable for some hidden Markov models with noncompact state spaces. It is known that, under some mild conditions, the error associated with particle approximation of filtering distributions satisfies a central limit theorem. The first stability property obtained by the author is a time-uniform bound on the corresponding asymptotic variance. The second stability property obtained is a linear-in-time …
description: scientific article; zbMATH DE number 6247418