LABS回归模型在贝叶斯框架下自适应逼近复杂函数，逼近速度接近最优。
Posterior Contraction of Lévy Adaptive B-spline Regression in Besov Spaces
- 用可变阶数和独立节点的B样条构建灵活的非参数回归模型
- 后验收缩率在贝叶斯空间中接近最小最大最优，仅差对数因子
- 适用于光滑性未知、结构不规则的函数建模，适合理论研究者
我们研究了莱维自适应B样条（LABS）回归模型的渐近性质，这是一种将B样条核引入莱维自适应回归核（LARK）模型的贝叶斯非参数方法。LABS采用不同阶数且独立定义节点的样条，形成能适应真实函数不规则与局部结构特征的灵活模型类。在单变量随机设计与高斯误差的非参数回归框架下，我们证明了LABS后验在贝叶斯空间中以近乎最小最大最优的速率收缩到真实函数，仅差一个对数因子，并能自动适应未知光滑性。该研究填补了文献空白，此前关于LARK模型在贝叶斯空间中的后验收缩理论极为有限。对贝叶斯空间中的标准测试函数（包括Blocks、Bumps、HeaviSine、Doppler）的模拟实验验证了理论结果，并展示了LABS的实际有效性。
原文摘要 · Abstract (English)
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.
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