Cauchy Markov Random Field Priors for Bayesian Inversion

Chada, Neil K.; Roininen, Lassi; Suuronen, Jarkko

doi:10.48550/arxiv.2105.12488

Cited by 3 publications

(3 citation statements)

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“…piecewise continuous or piecewise constant samples where the subdomains on which the sampled functions are continuous/constant are also random. Chada et al [6], for instance, discuss Cauchy random fields. It would be highly interesting to study such non-Gaussian random fields on, e.g., spaces of functions of bounded variation.…”

Section: Discussionmentioning

confidence: 99%

Gaussian random fields on non-separable Banach spaces

Korolev¹,

Latz²,

Schönlieb³

2022

Preprint

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We study Gaussian random fields on certain Banach spaces and investigate conditions for their existence. Our results apply inter alia to spaces of Radon measures and Hölder functions. In the former case, we are able to define Gaussian white noise on the space of measures directly, avoiding, e.g., an embedding into a negative-order Sobolev space. In the latter case, we demonstrate how Hölder regularity of the samples is controlled by that of the covariance kernel and, thus, show a connection to the Theorem of Kolmogorov-Chentsov.

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Section: Discussionmentioning

confidence: 99%

Gaussian random fields on non-separable Banach spaces

Korolev¹,

Latz²,

Schönlieb³

2022

Preprint

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“…We note that there is another class of 'Cauchy measures' in the literature, namely the class of stochastic processes with Cauchy-distributed increments, as used by e.g. Markkanen et al (2019) and Chada et al (2021).…”

Section: Application To Cauchy Measuresmentioning

confidence: 99%

Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

et al. 2021

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We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

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“…As noted in (Asar et al, 2020), a limiting case of the NIG distribution is the Cauchy distribution, the ngme package therefore also includes methods for Cauchy random fields. Cauchy random fields were also recently investigated by Chada et al (2021) for Bayesian inverse problems, an area where the SPDE approach previously has been used extensively (see, e.g., Roininen et al, 2014Roininen et al, , 2019. Bayesian methods for the non-Gaussian SPDE models were also recently investigated by (Walder and Hanks, 2020), who provided several examples of their use.…”

Section: Non-gaussian Modelsmentioning

confidence: 99%

The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

Lindgren¹,

Bolin²,

Rue³

2021

Preprint

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Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matérn covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations.

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Cauchy Markov Random Field Priors for Bayesian Inversion

Cited by 3 publications

References 0 publications

Gaussian random fields on non-separable Banach spaces

Gaussian random fields on non-separable Banach spaces

Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running

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