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.

翻译：本文讨论Radon-Nikodym导数的估计问题。该问题出现在多种应用场景中，例如协变量偏移自适应、似然比检验、互信息估计以及条件概率估计。为解决上述问题，我们在再生核希尔伯特空间中采用通用正则化框架。通过综合考虑导数的光滑性及估计空间容量，建立了相应正则化算法的收敛速率。该分析基于一般源条件与正则化Christoffel函数。研究还表明，任意特定点处的Radon-Nikodym导数重构均可达到高阶精度。数值模拟验证了理论结果。