  We consider kernel based learning methods for regression and analyze what
happens to the risk minimizer when new variables, statistically independent of
input and target variables, are added to the set of input variables; this
problem arises, for example, in the detection of causality relations between
two time series. We find that the risk minimizer remains unchanged if we
constrain the risk minimization to hypothesis spaces induced by suitable kernel
functions. We show that not all kernel induced hypothesis spaces enjoy this
property. We present sufficient conditions ensuring that the risk minimizer
does not change, and show that they hold for inhomogeneous polynomial and
Gaussian RBF kernels. We also provide examples of kernel induced hypothesis
spaces whose risk minimizer changes if independent variables are added as
input.
