Randal Douc

Comparison of resampling schemes for particle filtering

This contribution is devoted to the comparison of various resampling approaches that have been proposed in the literature on particle filtering. It is first shown using simple arguments that the so-called residual and stratified methods do yield an improvement over the basic multinomial resampling approach. A simple counter-example showing that this property does not hold true for systematic resampling is given. Finally, some results on the large-sample behavior of the simple bootstrap filter algorithm are given. In particular, a central limit theorem is established for the case where resampling is performed using the residual approach.

Adaptive importance sampling in general mixture classes

In this paper, we propose an adaptive algorithm that iteratively updates both the weights and component parameters of a mixture importance sampling density so as to optimise the performance of importance sampling, as measured by an entropy criterion. The method, called M-PMC, is shown to be applicable to a wide class of importance sampling densities, which includes in particular mixtures of multivariate Student t distributions. The performance of the proposed scheme is studied on both artificial and real examples, highlighting in particular the benefit of a novel Rao-Blackwellisation device which can be easily incorporated in the updating scheme.

Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime

An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.

LIMIT THEOREMS FOR WEIGHTED SAMPLES WITH APPLICATIONS TO SEQUENTIAL MONTE CARLO METHODS

In the last decade, sequential Monte Carlo methods (SMC) emerged as a key tool in computational statistics [see, e.g., Sequential Monte Carlo Methods in Practice (2001) Springer, New York, Monte Carlo Strategies in Scientific Computing (2001) Springer, New York, Complex Stochastic Systems (2001) 109-173]. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles, which are generated recursively. Despite many theoretical advances [see, e.g., J. Roy. Statist. Soc. Ser. B 63 (2001) 127-146, Ann. Statist. 33 (2005) 1983-2021, Feynman-Kac Formulae. Genealogical and Interacting Particle Systems with Applications (2004) Springer, Ann. Statist. 32 (2004) 2385-2411], the large-sample theory of these approximations remains a question of central interest. In this paper we establish a law of large numbers and a central limit theorem as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the transformations of the weighted sample used in the SMC algorithm preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state-space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies, including branching, resampling at randomly selected times, and so on.

Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models

This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algorithm relies on forgetting properties of the filtering dynamics and the quality of the estimates produced is investigated, both theoretically and via simulations.

Convergence of Adaptive Sampling Schemes

In the design of ecient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performances of a given kernel can clarify how adequate it is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is quite complex and most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sucient convergence conditions for a wide class of population Monte Carlo algorithms and show that Rao‐ Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions simply do not benefit from repeated updating.In the design of efficient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performance of a given simulation kernel can clarify a posteriori how adequate this kernel is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao--Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions do not benefit from repeated updating.

Sequential Monte Carlo smoothing for general state space hidden Markov models

Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles.

Consistency of the maximum likelihood estimator for general hidden Markov models

Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chains.

On the existence of some ARCH(∞) processes

A new sufficient condition for the existence of a stationary causal solution of an equation is provided. This condition allows us to consider coefficients with power-law decay, so that it can be applied to the so-called FIGARCH processes, whose existence is thus proved.

Long-term stability of sequential Monte Carlo methods under verifiable conditions

This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs with possibly non-compact state space. In addition, we derive a similar time uniform bound on the asymptotic L-p error. Importantly, our results hold for misspecified models; that is, we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an HMM.