约束插值的Kimeldorf-Wahba对应的推广
Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation
摘要：
本文将再生核希尔伯特空间（RKHS）中Bayes估计与最优插值之间的对应关系推广到线性不等式约束（如有界性、单调性或凸性）的情形。在无约束插值的情况下，已知给定数据插值的高斯过程（GP）的后验分布的均值是最小化与GP相关联的RKHS中的范数的最优插值函数。在有约束的情况下，我们证明了后验分布的最大后验概率（MAP）或众数是RKHS中的最优约束插值函数。因此，一般的对应关系是用MAP估计而不是后验分布的平均值来实现的。最后给出了一个数值例子来说明这一结果。
英文摘要：
In this paper, we extend the correspondence between Bayes' estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of linear inequality constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given data interpolation is known to be the optimal interpolation function minimizing the norm in the RKHS associated to the GP. In the constrained case, we prove that the Maximum A Posteriori (MAP) or Mode of the posterior distribution is the optimal constrained interpolation function in the RKHS. So, the general correspondence is achieved with the MAP estimator and not the mean of the posterior distribution. A numerical example is given to illustrate this last result.
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