Random tree Besov priors: Data-driven regularisation parameter selection

arXiv:2601.12957v2 Announce Type: replace Abstract: We develop a data-driven algorithm for automatically selecting the wavelet density parameters 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 \cite{Kekkonen2023}, 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.