Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns

Lasanen, Sari

doi:10.3934/ipi.2012.6.267

Cited by 31 publications

(30 citation statements)

References 22 publications

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“…• Otherwise, when ν > 0 satisfying (34) does not exist, results in Section 3.1 remain also valid when, at each iteration t, for a given value of σ (t) , we replace Λ by D(σ (t) ). However, there is a main difference with respect to the case when ν > 0, which is that µ depends on the value of the mixing variable σ (t) and hence can take different values along the iterations.…”

Section: Proposed Algorithmsmentioning

confidence: 99%

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An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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Abstract:In this paper, we are interested in Bayesian inverse problems where either the data fidelity term or the prior distribution is Gaussian or driven from a hierarchical Gaussian model. Generally, Markov chain Monte Carlo (MCMC) algorithms allow us to generate sets of samples that are employed to infer some relevant parameters of the underlying distributions. However, when the parameter space is high-dimensional, the performance of stochastic sampling algorithms is very sensitive to existing dependencies between parameters. In particular, this problem arises when one aims to sample from a high-dimensional Gaussian distribution whose covariance matrix does not present a simple structure. Another challenge is the design of Metropolis-Hastings proposals that make use of information about the local geometry of the target density in order to speed up the convergence and improve mixing properties in the parameter space, while not being too computationally expensive. These two contexts are mainly related to the presence of two heterogeneous sources of dependencies stemming either from the prior or the likelihood in the sense that the related covariance matrices cannot be diagonalized in the same basis. In this work, we address these two issues. Our contribution consists of adding auxiliary variables to the model in order to dissociate the two sources of dependencies. In the new augmented space, only one source of correlation remains directly related to the target parameters, the other sources of correlations being captured by the auxiliary variables. Experiments are conducted on two practical image restoration problems-namely the recovery of multichannel blurred images embedded in Gaussian noise and the recovery of signal corrupted by a mixed Gaussian noise. Experimental results indicate that adding the proposed auxiliary variables makes the sampling problem simpler since the new conditional distribution no longer contains highly heterogeneous correlations. Thus, the computational cost of each iteration of the Gibbs sampler is significantly reduced while ensuring good mixing properties.

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Section: Proposed Algorithmsmentioning

confidence: 99%

“…Non-Gaussian models arise in numerous applications in inverse problems [34][35][36][37]. In this context, the posterior distribution is non-Gaussian and does not generally follow a standard probability model.…”

Section: Designing Efficient Proposals In Mh Algorithmsmentioning

confidence: 99%

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

Marnissi

Chouzenoux

Benazza-Benyahia

et al. 2018

Entropy

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“…General development of Bayes' Theorem for inverse problems on function space, along the lines described here, may be found in [17,92]. The reader is also directed to the papers [61,62] for earlier related material and to [63][64][65] for recent developments. • Section 3.3.…”

Section: Bibliographic Notesmentioning

confidence: 99%

The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

141

222

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These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications involving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Hölder continuous functions. Bayes' theorem is derived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte Carlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.

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“…For the considered Gaussian priors, it is enough to show that the discrete priors converge to continuous priors in the discretization limit, and more specifically, we only need to show that the discrete covariance matrix converges to a continuous covariance in the discretization limit. This would guarantee the convergence of the posterior distributions, and hence discretization-invariance, as shown by Lasanen 2012 [4,5].…”

Section: Introductionmentioning

confidence: 95%

Cauchy difference priors for edge-preserving Bayesian inversion

Markkanen¹,

Roininen

Huttunen

et al. 2019

Journal of Inverse and Ill-Posed Problems

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We study Cauchy-distributed difference priors for edge-preserving Bayesian statistical inverse problems. On the contrary to the well-known total variation priors, one-dimensional Cauchy priors are non-Gaussian priors also in the discretization limit. Cauchy priors have independent and identically distributed increments. One-dimensional Cauchy and Gaussian random walks are special cases of Lévy α-stable random walks with α = 1 and α = 2, respectively. Both random walks can be written in closed-form, and as priors, they provide smoothing and edge-preserving properties. We briefly discuss also continuous and discrete Lévy α-stable random walks, and generalize the methodology to two-dimensional priors.We apply the developed algorithm to one-dimensional deconvolution and two-dimensional X-ray tomography problems. We compute conditional mean estimates with single-component Metropolis-Hastings and maximum a posteriori estimates with Gauss-Newton-type optimization method. We compare the proposed tomography reconstruction method to filtered back-projection estimate and conditional mean estimates with Gaussian and total variation priors.

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Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns

Cited by 31 publications

References 22 publications

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

An Auxiliary Variable Method for Markov Chain Monte Carlo Algorithms in High Dimension

The Bayesian Approach to Inverse Problems

Cauchy difference priors for edge-preserving Bayesian inversion

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