On the consistency of ensemble transform filter formulations

Reich, Sebastian; Shin, So Jin

doi:10.3934/jcd.2014.1.177

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…We present a continuation method to address two challenging issues in Bayesian inverse problems, that of overcoming the computational intractability of the posterior distribution, and scanning the transition distributions between the prior and the posterior. We consider the continuous update of the distribution alongside to the optimal transport approach addressed in a series of papers [23,25,29]. The continuation method introduced in [29], and used in [16], formulates the inversion as optimal control of information transport from the prior to posterior where the PDF is interpreted by the solution of a Liouville type equation with the assumption of available partial observations.…”

Section: Introductionmentioning

confidence: 99%

“…We consider the continuous update of the distribution alongside to the optimal transport approach addressed in a series of papers [23,25,29]. The continuation method introduced in [29], and used in [16], formulates the inversion as optimal control of information transport from the prior to posterior where the PDF is interpreted by the solution of a Liouville type equation with the assumption of available partial observations.…”

Section: Introductionmentioning

confidence: 99%

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A continuation method in Bayesian inference

Dia¹

2019

Preprint

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We present a continuation method that entails generating a sequence of transition probability density functions from the prior to the posterior in the context of Bayesian inference for parameter estimation problems. The characterization of transition distributions, by tempering the likelihood function, results in a homogeneous nonlinear partial integro-differential equation whose existence and uniqueness of solutions are addressed. The posterior probability distribution comes as the interpretation of the final state of the path of transition distributions. A computationally stable scaling domain for the likelihood is explored for the approximation of the expected deviance, where we manage to hold back all the evaluations of the forward predictive model at the prior stage. It follows the computational tractability of the posterior distribution and opens access to the posterior distribution for direct samplings. To get a solution formulation of the expected deviance, we derive a partial differential equation governing the moments generating function of the loglikelihood. We show also that a spectral formulation of the expected deviance can be obtained for low-dimensional problems under certain conditions. The computational efficiency of the proposed method is demonstrated through three differents numerical examples that focus on analyzing the computational bias generated by the method, assessing the continuation method in the Bayesian inference with non-Gaussian noise, and evaluating its ability to invert a multimodal parameter of interest.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A continuation method in Bayesian inference

Dia¹

2019

Preprint

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On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

Bishop

Moral

2023

Math. Control Signals Syst.

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The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman–Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter’s behaviour, e.g. it is signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman–Bucy filtering. We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

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A Continuation Method in Bayesian Inference

Dia

2023

SIAM/ASA J. Uncertainty Quantification

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On the consistency of ensemble transform filter formulations

Cited by 3 publications

References 21 publications

A continuation method in Bayesian inference

A continuation method in Bayesian inference

On the mathematical theory of ensemble (linear-Gaussian) Kalman–Bucy filtering

A Continuation Method in Bayesian Inference

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