  We study the problem of predicting the future, though only in the
probabilistic sense of estimating a future state of a time-varying probability
distribution. This is not only an interesting academic problem, but solving
this extrapolation problem also has many practical application, e.g. for
training classifiers that have to operate under time-varying conditions. Our
main contribution is a method for predicting the next step of the time-varying
distribution from a given sequence of sample sets from earlier time steps. For
this we rely on two recent machine learning techniques: embedding probability
distributions into a reproducing kernel Hilbert space, and learning operators
by vector-valued regression. We illustrate the working principles and the
practical usefulness of our method by experiments on synthetic and real data.
We also highlight an exemplary application: training a classifier in a domain
adaptation setting without having access to examples from the test time
distribution at training time.
