In multivariate spline regression, the number and locations of knots influence the performance and interpretability significantly. However, due to non-differentiability and varying dimensions, there is no desirable frequentist method to make inference on knots. In this article, we propose a fully Bayesian approach for knot inference in multivariate spline regression. The existing Bayesian method often uses BIC to calculate the posterior, but BIC is too liberal and it will heavily overestimate the knot number when the candidate model space is large. We specify a new prior on the knot number to take into account the complexity of the model space and derive an analytic formula in the normal model. In the non-normal cases, we utilize the extended Bayesian information criterion to approximate the posterior density. The samples are simulated in the space with differing dimensions via reversible jump Markov chain Monte Carlo. We apply the proposed method in knot inference and manifold denoising. Experiments demonstrate the splendid capability of the algorithm, especially in function fitting with jumping discontinuity.

翻译：在多元样条回归中，节点的数量与位置对模型性能与可解释性具有显著影响。然而，由于不可微性与维度变化特性，现有频率学派方法难以对节点进行有效统计推断。本文提出一种用于多元样条回归节点推断的完全贝叶斯方法。现有贝叶斯方法常采用BIC计算后验概率，但BIC准则过于宽松，当候选模型空间较大时会严重高估节点数量。我们通过设定考虑模型空间复杂度的节点数量新先验分布，在正态模型中推导出解析表达式。针对非正态情形，采用扩展贝叶斯信息准则近似后验密度。通过可逆跳转马尔可夫链蒙特卡洛方法在变维度空间中进行样本抽样。我们将所提方法应用于节点推断与流形去噪任务。实验结果表明该算法具有卓越性能，尤其在含跳跃间断点的函数拟合问题中表现突出。