Uniform convergence over time of a nested particle filtering scheme for
recursive parameter estimation in state–space Markov models
recursive parameter estimation in state–space Markov models
File(s)
Accepted version
Author(s)
Crisan, DO
Miguez, J
Type
Journal Article
Abstract
We analyse the performance of a recursive Monte Carlo method for
the Bayesian estimation of the static
parameters of a discrete–time state–space Markov model. The alg
orithm employs two layers of particle
filters to approximate the posterior probability distribution of the m
odel parameters. In particular, the
first layer yields an empirical distribution of samples on the paramete
r space, while the filters in the second
layer are auxiliary devices to approximate the (analytically intractab
le) likelihood of the parameters. This
approach relates the novel algorithm to the recent sequential Mo
nte Carlo square (SMC
2
) method, which
provides a
non-recursive
solution to the same problem. In this paper, we investigate the appr
oximation
of integrals of real bounded functions with respect to the poster
ior distribution of the system parameters.
Under assumptions related to the compactness of the parameter
support and the stability and continuity
of the sequence of posterior distributions for the state–space m
odel, we prove that the
L
p
norms of the
approximation errors vanish asymptotically (as the number of Mont
e Carlo samples generated by the
algorithm increases) and uniformly over time. We also prove that, un
der the same assumptions, the
proposed scheme can asymptotically identify the parameter values
for a class of models. We conclude the
paper with a numerical example that illustrates the uniform converg
ence results by exploring the accuracy
and stability of the proposed algorithm operating with long sequence
s of observations.
the Bayesian estimation of the static
parameters of a discrete–time state–space Markov model. The alg
orithm employs two layers of particle
filters to approximate the posterior probability distribution of the m
odel parameters. In particular, the
first layer yields an empirical distribution of samples on the paramete
r space, while the filters in the second
layer are auxiliary devices to approximate the (analytically intractab
le) likelihood of the parameters. This
approach relates the novel algorithm to the recent sequential Mo
nte Carlo square (SMC
2
) method, which
provides a
non-recursive
solution to the same problem. In this paper, we investigate the appr
oximation
of integrals of real bounded functions with respect to the poster
ior distribution of the system parameters.
Under assumptions related to the compactness of the parameter
support and the stability and continuity
of the sequence of posterior distributions for the state–space m
odel, we prove that the
L
p
norms of the
approximation errors vanish asymptotically (as the number of Mont
e Carlo samples generated by the
algorithm increases) and uniformly over time. We also prove that, un
der the same assumptions, the
proposed scheme can asymptotically identify the parameter values
for a class of models. We conclude the
paper with a numerical example that illustrates the uniform converg
ence results by exploring the accuracy
and stability of the proposed algorithm operating with long sequence
s of observations.
Date Issued
2017-11-17
Date Acceptance
2017-06-01
Citation
Advances in Applied Probability, 2017, 49 (4), pp.1170-1200
ISSN
1475-6064
Publisher
Applied Probability Trust
Start Page
1170
End Page
1200
Journal / Book Title
Advances in Applied Probability
Volume
49
Issue
4
Copyright Statement
© Applied Probability Trust 2017. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/N023781/1
Subjects
0102 Applied Mathematics
0104 Statistics
Statistics & Probability
Publication Status
Published