Distances and dependence measures with reproducing kernel Hilbert spaces

I discuss a family of distances on the probability measures based on positive definite kernels and the associated reproducing kernel Hilbert spaces (RKHS). It is shown that, with an appropriate choice of a RKHS, the mean of the kernel in RKHS with respect to a probability uniquely determines the probability. With RKHS of this characteristic property, a distance on the probabilities can be defined as the distances of the means. This type of distances provides straightforward estimators with finite sample, unlike some other well-known distances on probabilities. Some statistical asymptotic results on the estimators are discussed. It is also easy to derive a dependence measure based on the distance by considering the distance between the joint probability and the product of the marginals. I will discuss a normalized dependence measure with positive definite kernel, and show an interesting relation with the conventional chi-square divergence.