Two Metropolis-Hastings algorithms for posterior measures with non-Gaussian priors in infinite dimensions

TL;DR

This paper develops two Metropolis-Hastings algorithms tailored for sampling from complex, non-Gaussian prior measures in infinite-dimensional spaces, introducing Bessel-K priors that interpolate between known distributions and are suitable for sparse modeling.

Contribution

It introduces two novel Metropolis-Hastings algorithms for non-Gaussian priors in infinite dimensions and proposes Bessel-K priors for flexible, sparse modeling.

Findings

Algorithms successfully sample from Bessel-K priors

Numerical examples demonstrate effectiveness in density estimation and deconvolution

Bessel-K priors interpolate between gamma and Besov priors

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.