Sequential Monte Carlo smoothing for general state space hidden Markov models
type: Item
instance of: scholarly article
title: Sequential Monte Carlo smoothing for general state space hidden Markov models
zbMATH Open document ID: 1237.60026
DOI: 10.1214/10-AAP735
author: Randal Douc
author: Aurélien Garivier
author: Jimmy Olsson
author: Eric Moulines
published in: The Annals of Applied Probability
publication date: 2012-01-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.2945
review text: For any sequence \(\{a_n\}\), \(n\geq 0\), and any pair of integers \(0\leq m\leq n\), let \(a_{m:n}=(a_m,\ldots, a_n)\). All random variables are defined on a common probability space \((\Omega, \mathcal{F}, \mathrm{P})\), The sets \(\mathbb{X}\) and \(\mathbb{Y}\) are Polish spaces, \(B(\mathbb{X})\) and \(B(\mathbb{Y})\) are the associated Borel \(\sigma\)-algebras. The state process \(\{X_t\}_{t\geq 0}\) is a Markov chain on the state space \(\{\mathbb{X},B(\mathbb{X})\}\) with the initial distribution \(\chi\) and the transition kernel \(M\). The state process is assumed to be hidden but partially observed through the observation \(\{Y_t\}_{t}\), which are \(\mathbb{Y}\)-valued random variables conditionally independent given the latent state sequence \(\{X_t\}_{t\geq0}\). Let there exists a \(\sigma\)-finite measure \(\lambda\) on \(\{\mathbb{Y},B(\mathbb{Y})\}\) and a nonnegative transition density function \(g\) on \(\mathbb{X}\times\mathbb{Y}\) such that \[ \mathrm{P}\{Y_t\in A|X_t\}=\int\limits_{A}g(X_t,y)\lambda dy \text{ for all } A\in B(\mathbb{Y}). \] The kernel \(M\) and the transition destiny \(g\) are known. In this paper it is assumed that there exists arbitrary but fixed observations \(Y_{0:T}=\{y_0, \ldots, y_{T}\}\). The main task is the estimation of the posterior distribution of (different subsets of) the state vector \(X_{0:T}\) given these observations. Depending on the observations \(Y_{0:T}\), the state sequence \(\{X_s\}\), \(s\geq 0\), is a time inhomogeneous Markov chain. The aim of this paper is to provide a foundation of the particle-based approximation of such distributions and to analyse, in a common unifying framework, different schemes producing such approximations. The sequential Monte Carlo (SMC) method is applied as well. General convergence results, including exponential deviation inequalities and the central limit theorem (CLT) are obtained. The paper is organized as follows. The introduction contains the statements of the pr…
reviewed by: Nijolė Kalinauskaitė
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 5996107
zbMATH Keywords: sequential Monte Carlo methods
zbMATH Keywords: particle filter
zbMATH Keywords: smoothing
zbMATH Keywords: hidden Markov models
MaRDI profile type: Publication
arXiv ID: 1202.2945
cites work: On Approximate Maximum-Likelihood Methods for Blind Identification: How to Cope With the Curse of Dimensionality
cites work: Smoothing algorithms for state-space models
cites work: Inference in hidden Markov models.
cites work: Stability of nonlinear filters in nonmixing case
cites work: Q3015748
cites work: Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference
cites work: The simulation smoother for time series models
cites work: Q4819702
cites work: Particle methods: An introduction with applications
cites work: A backward particle interpretation of Feynman-Kac formulae
cites work: Q4501607
cites work: Q3723577
cites work: Forgetting the initial distribution for hidden Markov models
cites work: Limit theorems for weighted samples with applications to sequential Monte Carlo methods
cites work: On the auxiliary particle filter
cites work: Sequential Monte Carlo Methods in Practice
cites work: A sequential smoothing algorithm with linear computational cost
cites work: Monte Carlo Smoothing for Nonlinear Time Series
cites work: Monte Carlo techniques to estimate the conditional expectation in multi-stage non-linear filtering†
cites work: A note on auxiliary particle filters
cites work: Array algorithms for H/sup ∞/ estimation
cites work: On discrete time ergodic filters with wrong initial data
cites work: Recursive Monte Carlo filters: algorithms and theoretical analysis
cites work: Monte Carlo strategies in scientific computing
cites work: Filtering via Simulation: Auxiliary Particle Filters
cites work: Q4457489
cites work: Q3996430
cites work: Model robustness of finite state nonlinear filtering over the infinite time horizon
cites work: Discrete time nonlinear filters with informative observations are stable
cites work: Uniform observability of hidden Markov models and filter stability for unstable signals
cites work: The stability of conditional Markov processes and Markov chains in random environments
cites work: Uniform time average consistency of Monte Carlo particle filters
OpenAlex ID: W1965828084
description: scientific article