Bayesian estimation of incompletely observed diffusions

Meulen, Frank van der; Schauer, Moritz

doi:10.1080/17442508.2017.1381097

Cited by 15 publications

(34 citation statements)

References 24 publications

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“…The only thing that needs to be checked is thatρ(t, x) satisfies the Kolmogorov backward equation associated withX. This can be proved along the lines of Lemma 3.4 and Corollary 3.5 of [40]…”

Section: Proof Of Corollary 21mentioning

confidence: 96%

“…However, no simulation results were included in the paper, as 'these simulations proved prohibitively slow and the resulting method does not seem like a useful approach to sampling' [18, page 671]. We will shortly review in more detail the works [13], [27], and [40], as the present paper builds upon these. The first of these papers includes some forms of hypo-elliptic diffusions, whereas the latter two papers consider uniformly elliptic diffusions with L = I.…”

Section: Literature Reviewmentioning

confidence: 99%

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Simulation of elliptic and hypo-elliptic conditional diffusions

2020

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Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time T > 0 only linear combinations of its components are observed. That is, one only observes the vector LXT for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in (Schauer et al. (2017)), can be used in a unified way for both uniformly and hypo-elliptic diffusions, also when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.

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Section: Proof Of Corollary 21mentioning

confidence: 96%

Section: Literature Reviewmentioning

confidence: 99%

Simulation of elliptic and hypo-elliptic conditional diffusions

2020

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“…This could potentially relax the requirements on the extra drift term to only ensure convergence towards a given distance of the target. Inexact observations of stochastic processes are for example treated in [63].…”

Section: The Noise Fieldsmentioning

confidence: 99%

A Geometric Framework for Stochastic Shape Analysis

2018

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We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the large deformation diffeomorphic metric mapping framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finitedimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker-Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the FokkerPlanck equation to sample moments of the data, while the second approach employs an expectation-maximization based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.

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“…It remains to provide an effective design for g. One can use proposals developed for problems whereby a finite-dimensional SDE generates linear Gaussian observations and one is interested to perform a similar IS method, see e.g. [22,48,37,38,44].…”

Section: Likelihood-informed Proposalsmentioning

confidence: 99%

“…For the finite-dimensional SDE case more elaborate guiding functions can be found in [48,44] and some of these could be potentially extended so that they can be used in the SPDE setting instead of (16). The advantage of using g in (16) is that it provides a simple functional and can perform well for problems where t n − t n−1 is of moderate length, as also confirmed in the numerical examples of Section 5.…”

Section: Likelihood-informed Proposalsmentioning

confidence: 99%

Particle Filtering for Stochastic Navier--Stokes Signal Observed with Linear Additive Noise

Llopis¹,

Kantas²,

Beskos³

et al. 2018

SIAM J. Sci. Comput.

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We consider a non-linear filtering problem, whereby the signal obeys the stochastic Navier-Stokes equations and is observed through a linear mapping with additive noise. The setup is relevant to data assimilation for numerical weather prediction and climate modelling, where similar models are used for unknown ocean or wind velocities. We present a particle filtering methodology that uses likelihood informed importance proposals, adaptive tempering, and a small number of appropriate Markov Chain Monte Carlo steps. We provide a detailed design for each of these steps and show in our numerical examples that they are all crucial in terms of achieving good performance and efficiency.

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Bayesian estimation of incompletely observed diffusions

Cited by 15 publications

References 24 publications

Simulation of elliptic and hypo-elliptic conditional diffusions

Simulation of elliptic and hypo-elliptic conditional diffusions

A Geometric Framework for Stochastic Shape Analysis

Particle Filtering for Stochastic Navier--Stokes Signal Observed with Linear Additive Noise

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