Bayesian inversion with α-stable priors

Suuronen, Jarkko; Soto, Tomás; Chada, Neil K; Roininen, Lassi

doi:10.1088/1361-6420/acf154

Cited by 4 publications

(2 citation statements)

References 46 publications

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“…Infinite-dimensional Bayesian approaches guarantee the convergence to the continuous posterior and are independent of the discretization level. Despite the great success of such methods in modeling smooth parameters, addressing non-smooth or discontinuous parameters in the context of Bayesian inverse problems is still an ongoing direction of research (Suuronen et al, 2023(Suuronen et al, , 2022Uribe et al, 2022;Li et al, 2022).…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

Horst,

Afkham,

Dong

et al. 2024

Preprint

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In this work, we investigate the use of Besov priors in the context of Bayesian inverse problems. The solution to Bayesian inverse problems is the posterior distribution which naturally enables us to interpret the uncertainties. Besov priors are discretization invariant and can promote sparsity in terms of wavelet coefficients. We propose the randomize-then-optimize method to draw samples from the posterior distribution with Besov priors under a general parameter setting and estimate the modes of the posterior distribution. The performance of the proposed method is studied through numerical experiments of a 1D inpainting problem, a 1D deconvolution problem, and a 2D computed tomography problem. Further, we discuss the influence of the choice of the Besov parameters and the wavelet basis in detail, and we compare the proposed method with the state-of-the-art methods. The numerical results suggest that the proposed method is an effective tool for sampling the posterior distribution equipped with general Besov priors.

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Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

Horst,

Afkham,

Dong

et al. 2024

Preprint

View full text Add to dashboard Cite

show abstract

“…Other classical choices for heavy-tailed priors are α-stable distributions, such as the Cauchy distribution [4]. Heavy-tailed priors are especially popular in image reconstruction to preserve sharp edges.…”

Section: Introductionmentioning

confidence: 99%

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

Flock,

Dong,

Uribe

et al. 2023

Preprint

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We consider high-dimensional Bayesian inverse problems with arbitrary likelihood and product-form Laplace prior for which we provide a certified approximation of the posterior density in the Hellinger distance. The approximate posterior density differs from the prior density only in a small number of relevant coordinates that contribute the most to the update from the prior to the posterior. We propose and analyze a gradient-based diagnostic to identify these relevant coordinates. Although this diagnostic requires computing an expectation with respect to the posterior, we propose tractable methods for the classical case of a linear forward model with Gaussian likelihood. Our methods can be employed to estimate the diagnostic before solving the Bayesian inverse problem via, e.g., Markov chain Monte Carlo (MCMC) methods. After selecting the coordinates, the approximate posterior density can be efficiently inferred since most of its coordinates are only informed by the prior. Moreover, specialized MCMC methods, such as the pseudo-marginal MCMC algorithm, can be used to obtain less correlated samples when sampling the exact posterior density. We show the applicability of our method using a 1D signal deblurring problem and a high-dimensional 2D super-resolution problem.

show abstract

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

Flock,

Dong,

Uribe

et al. 2024

Stat Comput

View full text Add to dashboard Cite

We consider high-dimensional Bayesian inverse problems with arbitrary likelihood and product-form Laplace prior for which we provide a certified approximation of the posterior in the Hellinger distance. The approximate posterior differs from the prior only in a small number of relevant coordinates that contribute the most to the update from the prior to the posterior. We propose and analyze a gradient-based diagnostic to identify these relevant coordinates. Although this diagnostic requires computing an expectation with respect to the posterior, we propose tractable methods for the classical case of a linear forward model with Gaussian likelihood. Our methods can be employed to estimate the diagnostic before solving the Bayesian inverse problem via, e.g., Markov chain Monte Carlo (MCMC) methods. After selecting the coordinates, the approximate posterior can be efficiently inferred since most of its coordinates are only informed by the prior. Moreover, specialized MCMC methods, such as the pseudo-marginal MCMC algorithm, can be used to obtain less correlated samples when sampling the exact posterior. We show the applicability of our method using a 1D signal deblurring problem and a high-dimensional 2D super-resolution problem.

show abstract

Bayesian inversion with α-stable priors

Cited by 4 publications

References 46 publications

Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

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