Random tree Besov priors: Data-driven regularisation parameter selection

TL;DR

This paper introduces a hierarchical, data-driven method for selecting regularisation parameters in Bayesian inversion using random tree Besov priors, enhancing adaptivity and computational efficiency.

Contribution

It extends random tree Besov priors with a hierarchical model for automatic, data-driven regularisation parameter selection across scales.

Findings

Effective in nonparametric regression

Preliminary promising results in deconvolution

Maintains computational efficiency

Abstract

We develop a data-driven algorithm for automatically selecting the regularisation parameter in Bayesian inversion under random tree Besov priors. One of the key challenges in Bayesian inversion is the construction of priors that are both expressive and computationally feasible. Random tree Besov priors, introduced in Kekkonen et al. (2023), provide a flexible framework for capturing local regularity properties and sparsity patterns in a wavelet basis. In this paper, we extend this approach by introducing a hierarchical model that enables data-driven selection of the wavelet density parameter, allowing the regularisation strength to adapt across scales while retaining computational efficiency. We focus on nonparametric regression and also present preliminary plug-and-play results for a deconvolution problem.