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. Although both methods are widely employed, they each have 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. An application to a real dataset substantiates the validity of our proposed approach.

翻译：贝叶斯P样条和基于贝叶斯模型选择的基确定策略，是贝叶斯框架下利用样条基展开进行非参数回归的两种常用方法。尽管这两种方法被广泛采用，但它们各自存在特定局限性，可能因目标函数的性质不同而导致潜在估计偏差。为克服各自方法的局限性并发挥其优势，我们提出了一种整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构成的惩罚项来评估样条模型的复杂度。所提方法能够自适应未知平滑度水平，同时在对数因子范围内达到极小极大最优后验收缩率。我们构建了高效的马尔可夫链蒙特卡洛算法以实现该方法。广泛的仿真研究表明，所提方法在性能指标或模型复杂度方面优于其他竞争方法。真实数据集的应用验证了该方法的有效性。