Edge preserving priors for inverse problems
Conference
65th ISI World Statistics Congress 2025
Format: IPS Abstract - WSC 2025
Keywords: "bayesian, prior, sparsity
Session: IPS 1024 - Bayesian Inverse Problems
Monday 6 October 10:50 a.m. - 12:30 p.m. (Europe/Amsterdam)
Abstract
The Bayesian approach to inverse problems allows us to encode our a priori knowledge of the unknown function of interest as a probability distribution. Gaussian process priors are often used in Bayesian inverse problems due to their fast computational properties. However, the smoothness of the resulting estimates is not well suited for modelling functions with sharp changes, such as images. Smooth functions with few local irregularities have a sparse expansion in the wavelet basis, making wavelet-based Besov priors a good candidate for modelling spatially inhomogeneous functions. The sparsity-promoting and edge-preservation properties of Besov priors can be further enhanced by introducing a new random variable that takes values in the space of ‘trees,’ ensuring that the realisations have jumps only on a small set. We will also discuss how to estimate the optimal value for the hyperparameter controlling the sparsity of the solution from the data.