Classification and Multiple Testing

This chapter considers two fundamental inference questions arising in the study of sparse high-dimensional models. In this setting, the true parameter typically only has a few non-zero coordinates. An important question is that of determining simultaneously for all coordinates whether they are zero or not. We introduce typical losses for this problem: the classification loss, as well as the False Discovery rate and False Negative rate. We determine the sharp minimax risks in this setting, and discuss a natural way to achieve these optimal bounds by using a decision-theoretic motivated Bayesian method, that simply rejects the null hypothesis if the posterior probability of the signal being zero is small enough.