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Published October 8, 2019 | Published + Submitted
Journal Article Open

Two Metropolis-Hastings Algorithms for Posterior Measures with Non-Gaussian Priors in Infinite Dimensions

Abstract

We introduce two classes of Metropolis--Hastings algorithms for sampling target measures that are absolutely continuous with respect to non-Gaussian prior measures on infinite-dimensional Hilbert spaces. In particular, we focus on certain classes of prior measures for which prior-reversible proposal kernels of the autoregressive type can be designed. We then use these proposal kernels to design algorithms that satisfy detailed balance with respect to the target measures. Afterwards, we introduce a new class of prior measures, called the Bessel-K priors, as a generalization of the gamma distribution to measures in infinite dimensions. The Bessel-K priors interpolate between well-known priors such as the gamma distribution and Besov priors and can model sparse or compressible parameters. We present concrete instances of our algorithms for the Bessel-K priors in the context of numerical examples in density estimation, finite-dimensional denoising, and deconvolution on the circle.

Additional Information

© 2019 by SIAM and ASA. Received by the editors April 25, 2018; accepted for publication (in revised form) May 15, 2019; published electronically October 8, 2019. The author is thankful to Profs. Derek Bingham, David Campbell, Nilima Nigam, and Andrew M. Stuart as well as Dr. James E. Johndrow and Sam Powers for interesting discussions and comments. We also owe a debt of gratitude to the anonymous reviewers whose careful comments and questions helped us improve an earlier version of this article. This work was supported by a PDF fellowship granted by the Natural Sciences and Engineering Research Council of Canada.

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Submitted - 1804.07833.pdf

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August 19, 2023
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