Posterior Contraction of L\'evy Adaptive B-spline Regression in Besov Spaces

TL;DR

This paper proves that the Lévý Adaptive B-spline regression model's Bayesian posterior concentrates around the true function at nearly optimal rates in Besov spaces, demonstrating both theoretical rigor and practical effectiveness.

Contribution

It provides the first theoretical analysis of posterior contraction rates for the LARK model in Besov spaces, showing LABS's adaptivity and near-minimax optimality.

Findings

Posterior contracts at nearly minimax rates in Besov spaces.

LABS adapts automatically to unknown smoothness.

Simulation results validate theoretical findings.

Abstract

We investigate the asymptotic properties of the L\'evy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the L\'evy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.