Two Metropolis-Hastings algorithms for posterior measures with non-Gaussian priors in infinite dimensions
This work provides novel sampling algorithms for Bayesian inverse problems with non-Gaussian priors, addressing a known bottleneck in infinite-dimensional inference.
The paper introduces two Metropolis-Hastings algorithms for sampling posterior measures with non-Gaussian priors in infinite dimensions, specifically for Bessel-K priors, and demonstrates their effectiveness in density estimation, denoising, and deconvolution tasks.
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.