Error Bounds and Normalising Constants for Sequential Monte Carlo Samplers in High Dimensions

Beskos, Alexandros; Crisan, Dan; Jasra, Ajay; Whiteley, Nick

doi:10.1017/s0001867800007047

Cited by 22 publications

(34 citation statements)

References 27 publications

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“…This curse of dimensionality does not affect most classical tracking problems, whose dimension is typically of order unity, but becomes absolutely prohibitive in large-scale data assimilation problems such as weather forecasting where model dimensions of order 10 7 are routinely encountered [1]. While the curse of dimensionality problem in particle filters is now fairly well understood, there exists no rigorous approach to date for alleviating this problem [2,15,19]. Practical data assimilation in high-dimensional models is therefore generally performed by means of ad-hoc algorithms, frequently based on (questionable) Gaussian approximations, that possess limited theoretical justification [1,9,11].…”

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confidence: 99%

Can local particle filters beat the curse of dimensionality?

Rebeschini¹,

Handel²

2015

Ann. Appl. Probab.

181

262

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The discovery of particle filtering methods has enabled the use of nonlinear filtering in a wide array of applications. Unfortunately, the approximation error of particle filters typically grows exponentially in the dimension of the underlying model. This phenomenon has rendered particle filters of limited use in complex data assimilation problems. In this paper, we argue that it is often possible, at least in principle, to develop local particle filtering algorithms whose approximation error is dimension-free. The key to such developments is the decay of correlations property, which is a spatial counterpart of the much better understood stability property of nonlinear filters. For the simplest possible algorithm of this type, our results provide under suitable assumptions an approximation error bound that is uniform both in time and in the model dimension. More broadly, our results provide a framework for the investigation of filtering problems and algorithms in high dimension.Comment: Published at http://dx.doi.org/10.1214/14-AAP1061 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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confidence: 99%

Can local particle filters beat the curse of dimensionality?

Rebeschini¹,

Handel²

2015

Ann. Appl. Probab.

181

262

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“…The issue of dimensionality in SMC methods has attracted substantial attention in the literature [2,3,4,24]. In this section, using a simple approach for the analysis of particle filters which is clearly exposed in [24], we show that for our SMC method it is possible to prove dimension-free error bounds.…”

Section: Convergence Propertymentioning

confidence: 84%

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

et al. 2015

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In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic SMC method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the sequential Monte Carlo (SMC) methods for inverse problems which were introduced in [20]; the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology, and its desirable theoretical properties, are demonstrated on numerical examples in both two and three dimensions.

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“…3 Sequential Monte Carlo approaches SMC samplers (Del Moral et al 2006) are a generalisation of IS, in which the problem of choosing an appropriate proposal distribution in IS is avoided by performing IS sequentially on a sequence of target distributions, starting at a target that is easy to simulate from, and ending at the target of interest. In standard IS the number of Monte Carlo points required in order to obtain a particular accuracy increases exponentially with the dimension of the space, but Beskos et al (2011) show (under appropriate regularity conditions) that the use of SMC circumvents this problem and can thus be practically useful in high dimensions.…”

Section: Discussionmentioning

confidence: 99%

Bayesian model comparison with un-normalised likelihoods

et al. 2016

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Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalising constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes' factors (BFs) for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates

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Error Bounds and Normalising Constants for Sequential Monte Carlo Samplers in High Dimensions

Cited by 22 publications

References 27 publications

Can local particle filters beat the curse of dimensionality?

Can local particle filters beat the curse of dimensionality?

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

Bayesian model comparison with un-normalised likelihoods

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