On regularized Radon-Nikodym differentiation
Duc Hoan Nguyen Werner Zellinger Sergei V. Pereverzyev
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.