arXiv subjects

Jimmy Olsson

Publications and source records attributed to Jimmy Olsson.

47 records · Page 3Linked to original sources

arXiv2014

Comparison of asymptotic variances of inhomogeneous Markov chains with application to Markov chain Monte Carlo methods

In this paper, we study the asymptotic variance of sample path averages for inhomogeneous Markov chains that evolve alternatingly according to two different $π$-reversible Markov transition kernels $P$ and $Q$. More specifically, our main result allows us to compare directly the asymptotic variances of two inhomogeneous Markov chains associated with different kernels $P_i$ and $Q_i$, $i\in\{0,1\}$, as soon as the kernels of each pair $(P_0,P_1)$ and $(Q_0,Q_1)$ can be ordered in the sense of lag-one autocovariance. As an important application, we use this result for comparing different data-augmentation-type Metropolis-Hastings algorithms. In particular, we compare some pseudo-marginal algorithms and propose a novel exact algorithm, referred to as the random refreshment algorithm, which is more efficient, in terms of asymptotic variance, than the Grouped Independence Metropolis-Hastings algorithm and has a computational complexity that does not exceed that of the Monte Carlo Within Metropolis algorithm.

stat.ME↗

arXiv2014

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 noncompact state space. In addition, we derive a similar time uniform bound on the asymptotic $\mathsf{L}^p$ 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.

math.ST↗

arXiv2014

On the use of Markov chain Monte Carlo methods for the sampling of mixture models

In this paper we study asymptotic properties of different data-augmentation-type Markov chain Monte Carlo algorithms sampling from mixture models comprising discrete as well as continuous random variables. Of particular interest to us is the situation where sampling from the conditional distribution of the continuous component given the discrete component is infeasible. In this context, we cast Carlin & Chib's pseudo-prior method into the framework of mixture models and discuss and compare different variants of this scheme. We propose a novel algorithm, the FCC sampler, which is less computationally demanding than any Metropolised Carlin & Chib-type algorithm. The significant gain of computational efficiency is however obtained at the cost of some asymptotic variance. The performance of the algorithm vis-à-vis alternative schemes is investigated theoretically, using some recent results obtained in [3] for inhomogeneous Markov chains evolving alternatingly according to two different reversible Markov transition kernels, as well as numerically.

stat.CO↗

arXiv2012

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.

math.PR↗

arXiv2011

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.

math.ST↗

arXiv2010

Metropolising forward particle filtering backward sampling and Rao-Blackwellisation of Metropolised particle smoothers

Smoothing in state-space models amounts to computing the conditional distribution of the latent state trajectory, given observations, or expectations of functionals of the state trajectory with respect to this distributions. For models that are not linear Gaussian or possess finite state space, smoothing distributions are in general infeasible to compute as they involve intergrals over a space of dimensionality at least equal to the number of observations. Recent years have seen an increased interest in Monte Carlo-based methods for smoothing, often involving particle filters. One such method is to approximate filter distributions with a particle filter, and then to simulate backwards on the trellis of particles using a backward kernel. We show that by supplementing this procedure with a Metropolis-Hastings step deciding whether to accept a proposed trajectory or not, one obtains a Markov chain Monte Carlo scheme whose stationary distribution is the exact smoothing distribution. We also show that in this procedure, backward sampling can be replaced by backward smoothing, which effectively means averaging over all possible trajectories. In an example we compare these approaches to a similar one recently proposed by Andrieu, Doucet and Holenstein, and show that the new methods can be more efficient in terms of precision (inverse variance) per computation time.

stat.CO↗

arXiv2010

Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators

This paper concerns the use of the expectation-maximisation (EM) algorithm for inference in partially observed diffusion processes. In this context, a well known problem is that all except a few diffusion processes lack closed-form expressions of the transition densities. Thus, in order to estimate efficiently the EM intermediate quantity we construct, using novel techniques for unbiased estimation of diffusion transition densities, a random weight fixed-lag auxiliary particle smoother, which avoids the well known problem of particle trajectory degeneracy in the smoothing mode. The estimator is justified theoretically and demonstrated on a simulated example.

math.ST↗

arXiv2009

On the Forward Filtering Backward Smoothing particle approximations of the smoothing distribution in general state spaces models

A prevalent problem in general state-space models is the approximation of the smoothing distribution of a state, or a sequence of states, conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous foundation for the calculation, or approximation, of such smoothed distributions, and to analyse in a common unifying framework different schemes to reach this goal. Through a cohesive and generic exposition of the scientific literature we offer several novel extensions allowing to approximate joint smoothing distribution in the most general case with a cost growing linearly with the number of particles.

math.ST↗

arXiv2008

Adaptive methods for sequential importance sampling with application to state space models

In this paper we discuss new adaptive proposal strategies for sequential Monte Carlo algorithms--also known as particle filters--relying on criteria evaluating the quality of the proposed particles. The choice of the proposal distribution is a major concern and can dramatically influence the quality of the estimates. Thus, we show how the long-used coefficient of variation of the weights can be used for estimating the chi-square distance between the target and instrumental distributions of the auxiliary particle filter. As a by-product of this analysis we obtain an auxiliary adjustment multiplier weight type for which this chi-square distance is minimal. Moreover, we establish an empirical estimate of linear complexity of the Kullback-Leibler divergence between the involved distributions. Guided by these results, we discuss adaptive designing of the particle filter proposal distribution and illustrate the methods on a numerical example.

stat.CO↗

arXiv2008

Sequential Monte Carlo smoothing with application to parameter estimation in non-linear 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.

math.ST↗

arXiv2007

On the auxiliary particle filter

In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) 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. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.

math.ST↗