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Cauchy Markov random field priors for Bayesian inversion

Abstract

The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed. In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo methods are usually required. In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison. We firstly propose a one-dimensional second-order Cauchy difference prior, and construct new first- and second-order two-dimensional isotropic Cauchy difference priors. Another new Cauchy prior is based on the stochastic partial differential equation approach, derived from Matérn type Gaussian presentation. The comparison also includes Cauchy sheets. Our numerical computations are based on both maximum a posteriori and conditional mean estimation. We exploit state-of-the-art MCMC methodologies such as Metropolis-within-Gibbs, Repelling-Attracting Metropolis, and No-U-Turn sampler variant of Hamiltonian Monte Carlo. We demonstrate the models and methods constructed for one-dimensional and two-dimensional deconvolution problems. Thorough MCMC statistics are provided for all test cases, including potential scale reduction factors.

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Acknowledgements

The authors thank Dr Sari Lasanen and Prof. Heikki Haario for useful discussions. JS and LR acknowledge Academy of Finland project funding (grant numbers 334816 and 336787). NKC is supported by KAUST baseline funding.

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Correspondence to Neil K. Chada.

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Suuronen, J., Chada, N.K. & Roininen, L. Cauchy Markov random field priors for Bayesian inversion. Stat Comput 32, 33 (2022). https://doi.org/10.1007/s11222-022-10089-z

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  • DOI: https://doi.org/10.1007/s11222-022-10089-z

Keywords

  • Cauchy priors
  • Markov models
  • inverse problems
  • Bayesian approach
  • Metropolis algorithms
  • HMC-NUTS

AMS subject classifications

  • 62F15
  • 60G52
  • 60J22
  • 65C40