Bayesian P-splines and basis determination through Bayesian model selection are both commonly employed strategies for nonparametric regression using spline basis expansions within the Bayesian framework. Despite their widespread use, each method has particular limitations that may introduce potential estimation bias depending on the nature of the target function. To overcome the limitations associated with each method while capitalizing on their respective strengths, we propose a new prior distribution that integrates the essentials of both approaches. The proposed prior distribution assesses the complexity of the spline model based on a penalty term formed by a convex combination of the penalties from both methods. The proposed method exhibits adaptability to the unknown level of smoothness, while achieving the minimax-optimal posterior contraction rate up to a logarithmic factor. We provide an efficient Markov chain Monte Carlo algorithm for implementing the proposed approach. Our extensive simulation study reveals that the proposed method outperforms other competitors in terms of performance metrics or model complexity.

翻译：贝叶斯P-样条与通过贝叶斯模型选择进行基确定，都是贝叶斯框架下使用样条基展开进行非参数回归的常用策略。尽管应用广泛，每种方法均存在特定局限性，可能根据目标函数的性质引入潜在估计偏差。为克服各方法的局限性并发挥其各自优势，我们提出一种融合两种方法核心要素的新先验分布。该先验分布基于两种方法惩罚项的凸组合形成的惩罚项来评估样条模型的复杂度。所提方法能自适应未知平滑度，同时在对数因子范围内达到极小极大最优后验收缩率。我们为该方法实现了高效马尔可夫链蒙特卡洛算法。大量仿真研究表明，在性能指标或模型复杂度方面，所提方法优于其他竞争方法。