Hierarchical Besov-Laplace priors for spatially inhomogeneous binary classification

TL;DR

This paper introduces a hierarchical Bayesian approach using Besov-Laplace priors for spatially inhomogeneous binary classification, achieving optimal posterior concentration and demonstrating practical efficiency through MCMC algorithms and numerical tests.

Contribution

It develops a novel hierarchical Bayesian method with adaptive regularity tuning for binary classification, improving prior flexibility and computational efficiency.

Findings

Posterior concentrates at the optimal rate, adapting to unknown regularity.

The MCMC algorithm efficiently implements the Bayesian procedure.

Numerical simulations confirm the theoretical advantages.

Abstract

We study nonparametric Bayesian binary classification, in the case where the unknown probability response function is possibly spatially inhomogeneous, for example, being generally flat across the domain but presenting localized sharp variations. We consider a hierarchical procedure based on the popular Besov-Laplace priors from inverse problems and imaging, with a carefully tuned hyper-prior on the regularity parameter. We show that the resulting posterior distribution concentrates towards the ground truth at optimal rate, automatically adapting to the unknown regularity. To implement posterior inference in practice, we devise an efficient Markov chain Monte Carlo (MCMC) algorithm based on recent ad-hoc dimension-robust methods for Besov-Laplace priors. We then test the considered approach in extensive numerical simulations, where we obtain a solid corroboration of the theoretical results.