Multilevel particle filters: normalizing constant estimation

Jasra, Ajay; Kamatani, Kengo; Osei, Prince Peprah; Zhou, Yan

doi:10.1007/s11222-016-9715-5

Cited by 28 publications

(59 citation statements)

References 13 publications

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“…The requirement to of independent (or exact) sampling from couples with the correct marginals is often not possible in many contexts. This has been dealt with in several recent works, such as [2,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

A Multi-Index Markov Chain Monte Carlo Method

Jasra

Kamatani

Law

et al. 2018

Int. J. UncertaintyQuantification

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In this article we consider computing expectations w.r.t. probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space-time.In such contexts, it is known that the multi-index Monte Carlo (MIMC) method of [7] can improve upon i.i.d. sampling from the most accurate approximation of the probability law. Through a non-trivial modification of the multilevel Monte Carlo (MLMC) method, this method can reduce the work to obtain a given level of error, relative to i.i.d. sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to be sampled independently, for example a Bayesian inverse problem where evaluation of the likelihood requires solution of a partial differential equation (PDE) model which needs to be approximated at finite resolution. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem for a simplified estimator which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d. sampling from the most accurate approximation, under appropriate assumptions. The method is numerically illustrated on a Bayesian inverse problem associated to a stochastic partial differential equation (SPDE), where the path measure is conditioned on some observations.

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Section: Introductionmentioning

confidence: 99%

A Multi-Index Markov Chain Monte Carlo Method

Jasra

Kamatani

Law

et al. 2018

Int. J. UncertaintyQuantification

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“…We conclude this section with a technical theorem. We consider onlyη ML,L n (f ), but this can be extended toζ ML,L n (f ), similarly to [19] . The proofs are given in Appendix A.…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…We will describe the particle filter that is capable of exactly approximating, that is as the Monte Carlo samples go to infinity, terms of the form (19) and (20), for any fixed l. The particle filter has been studied and used extensively (see for example [5,8]) in many practical applications of interest.…”

Section: Particle Filteringmentioning

confidence: 99%

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Multilevel particle filters for Lévy-driven stochastic differential equations

2018

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We develop algorithms for computing expectations with respect to the laws of models associated to stochastic differential equations (SDEs) driven by pure Lévy processes. We consider filtering such processes and well as pricing of path dependent options. We propose a multilevel particle filter (MLPF) to address the computational issues involved in solving these continuum problems. We show via numerical simulations and theoretical results that under suitable assumptions regarding the discretization of the underlying driving Lévy proccess, our proposed method achieves optimal convergence rates: the cost to obtain MSE O( 2 ) scales like O( −2 ) for our method, as compared with the standard particle filter O( −3 ).

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“…(ii) The method can be used for approximating the expectation of some functionals w.r.t. the smoother, whereas the approach in [20,21] is typically not useful for smoothing at large time-lags. In this article we establish that (i) can hold in an ideal special case, where the model is linear and Gaussian and the transport map is exact.…”

mentioning

confidence: 99%

Multilevel Monte Carlo for Smoothing via Transport Methods

Houssineau¹,

Jasra²,

Singh³

2018

SIAM J. Sci. Comput.

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In this article we consider recursive approximations of the smoothing distribution associated to partially observed stochastic differential equations (SDEs), which are observed discretely in time. Such models appear in a wide variety of applications including econometrics, finance and engineering. This problem is notoriously challenging, as the smoother is not available analytically and hence require numerical approximation. This usually consists by applying a time-discretization to the SDE, for instance the Euler method, and then applying a numerical (e.g. Monte Carlo) method to approximate the smoother. This has lead to a vast literature on methodology for solving such problems, perhaps the most popular of which is based upon the particle filter (PF) e.g. [9]. In the context of filtering for this class of problems, it is well-known that the particle filter can be improved upon in terms of cost to achieve a given mean squared error (MSE) for estimates. This in the sense that the computational effort can be reduced to achieve this target MSE, by using multilevel (ML) methods [12,13,18], via the multilevel particle filter (MLPF) [16,20,21]. For instance, to obtain a MSE of O( 2 ) for some > 0 when approximating filtering distributions associated with Euler-discretized diffusions with constant diffusion coefficients, the cost of the PF is O( −3 ) while the cost of the MLPF is O( −2 log( ) 2 ). In this article we consider a new approach to replace the particle filter, using transport methods in [27]. In the context of filtering, one expects that the proposed method improves upon the MLPF by yielding, under assumptions, a MSE of O( 2 ) for a cost of O( −2 ). This is established theoretically in an "ideal" example and numerically in numerous examples.

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Multilevel particle filters: normalizing constant estimation

Cited by 28 publications

References 13 publications

A Multi-Index Markov Chain Monte Carlo Method

A Multi-Index Markov Chain Monte Carlo Method

Multilevel particle filters for Lévy-driven stochastic differential equations

Multilevel Monte Carlo for Smoothing via Transport Methods

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