Posterior Contraction of Lévy Adaptive B-spline Regression in Besov Spaces
Provides theoretical guarantees for a flexible Bayesian nonparametric regression method, filling a gap in the literature on posterior contraction of LARK models in Besov spaces.
The paper establishes that the Lévy Adaptive B-spline (LABS) regression model achieves nearly minimax-optimal posterior contraction rates in Besov spaces, adapting to unknown smoothness. Simulation experiments on standard test functions confirm its practical utility.
We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy 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.