On Bounds and Asymptotics of Sequential Monte Carlo Methods for Filtering,
Smoothing, and Maximum Likelihood Estimation in State Space Models
Centre for Mathematical Sciences
Lund Institute of Technology
This thesis is based on four papers (A-D) treating filtering, smoothing,
and maximum likelihood (ML) estimation in general state space models using
stochastic particle filters (also referred to as sequential Monte Carlo (SMC)
The aim of Paper A is to study the bias of Monte Carlo integration estimates
produced by the so-called bootstrap particle filter. A bound on this bias
which is inversely proportional to the number N of particles of the system
is established. In addition, we refine the analysis by deriving the asymptotic
bias as N tends to infinity and, under suitable mixing assumptions on the
latent Markov model, a time uniform bound.
In Paper B we consider ML estimation based on EM (Expectation-Maximization)
methods. In this context, the key ingredient is the computation of smoothed
sum functionals of the hidden states for given values of the model parameters.
It has been observed by several authors that using standard SMC methods for
this smoothing assignment may be unreliable for larger observations sizes.
Thus we study a simple variant, based on forgetting ideas of the state space
model dynamics, of the basic sequential smoothing approach which is transparent
in terms of computation time and reduces the variability of the sum functional
approximation. Under suitable regularity assumptions, it is shown that this
modification indeed allows a tighter control of the Lp error and the bias
of the approximation.
To perform ML estimation in state space models, the log-likelihood function
must be approximated. In Paper C we study such approximations based on particle
filters, and in particular conditions for consistency and asymptotic normality
of the corresponding approximate ML estimators. Numerical results illustrate
Paper D is devoted to the study asymptotic properties of weighted particle
samples produced by the so-called two-stage sampling (TSS) particle filter,
which is a generalization of the auxiliary particle filter proposed by Pitt
and Shephard (1999). Besides establishing a central limit theorem (CLT) for
smoothed particle estimates, we also derive bounds on the Lp error and bias
of the same for a finite particle sample size. Setting out from the recursive
formula for the asymptotic variance of the CLT, we discuss some possible
improvements of the TSS algorithm.
Asymptotic normality, consistency, EM algorithm, maximum likelihood, particle
filter, sequential Monte Carlo, smoothing, state space models, two-stage