We discuss the problem of estimating Radon-Nikodym derivatives. This problem appears in various applications, such as covariate shift adaptation, likelihood-ratio testing, mutual information estimation, and conditional probability estimation. To address the above problem, we employ the general regularization scheme in reproducing kernel Hilbert spaces. The convergence rate of the corresponding regularized algorithm is established by taking into account both the smoothness of the derivative and the capacity of the space in which it is estimated. This is done in terms of general source conditions and the regularized Christoffel functions. We also find that the reconstruction of Radon-Nikodym derivatives at any particular point can be done with high order of accuracy. Our theoretical results are illustrated by numerical simulations.

本文讨论了估计Radon-Nikodym导数的问题，在几个应用领域中出现，比如协变量转移适应，似然比检验，互信息估计和条件概率估计。我们采用了重构核希尔伯特空间中的一般正则化方案来解决上述问题，并通过考虑导数的平滑性和估计空间的容量来建立相应正则化算法的收敛速度，我们还发现在任何特定点上对Radon-Nikodym导数的重建可以以高阶精度进行。数值模拟结果验证了我们的理论结果。

关于正则化的Radon-Nikodym微分