Bayesian inversion with Student's t priors based on Gaussian scale mixtures
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913431178706944 |
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| author | Senchukova, Angelina Uribe, Felipe Roininen, Lassi |
| author_facet | Senchukova, Angelina Uribe, Felipe Roininen, Lassi |
| contents | Many inverse problems focus on recovering a quantity of interest that is a priori known to exhibit either discontinuous or smooth behavior. Within the Bayesian approach to inverse problems, such structural information can be encoded using Markov random field priors. We propose a class of priors that combine Markov random field structure with Student's t distribution. This approach offers flexibility in modeling diverse structural behaviors depending on available data. Flexibility is achieved by including the degrees of freedom parameter of Student's t distribution in the formulation of the Bayesian inverse problem. To facilitate posterior computations, we employ Gaussian scale mixture representation for the Student's t Markov random field prior, which allows expressing the prior as a conditionally Gaussian distribution depending on auxiliary hyperparameters. Adopting this representation, we can derive most of the posterior conditional distributions in a closed form and utilize the Gibbs sampler to explore the posterior. We illustrate the method with two numerical examples: signal deconvolution and image deblurring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13665 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bayesian inversion with Student's t priors based on Gaussian scale mixtures Senchukova, Angelina Uribe, Felipe Roininen, Lassi Computation Many inverse problems focus on recovering a quantity of interest that is a priori known to exhibit either discontinuous or smooth behavior. Within the Bayesian approach to inverse problems, such structural information can be encoded using Markov random field priors. We propose a class of priors that combine Markov random field structure with Student's t distribution. This approach offers flexibility in modeling diverse structural behaviors depending on available data. Flexibility is achieved by including the degrees of freedom parameter of Student's t distribution in the formulation of the Bayesian inverse problem. To facilitate posterior computations, we employ Gaussian scale mixture representation for the Student's t Markov random field prior, which allows expressing the prior as a conditionally Gaussian distribution depending on auxiliary hyperparameters. Adopting this representation, we can derive most of the posterior conditional distributions in a closed form and utilize the Gibbs sampler to explore the posterior. We illustrate the method with two numerical examples: signal deconvolution and image deblurring. |
| title | Bayesian inversion with Student's t priors based on Gaussian scale mixtures |
| topic | Computation |
| url | https://arxiv.org/abs/2403.13665 |