Heavy-tailed distributions are frequently used to enhance the robustness of
regression and classification methods to outliers in output space. Often,
however, we are confronted with ``outliers'' in input space, which are isolated
observations in sparsely populated regions. We show that heavy-tailed
stochastic processes (which we construct from Gaussian processes via a copula),
can be used to improve robustness of regression and classification estimators
to such outliers by selectively shrinking them more strongly in sparse regions
than in dense regions.