More Powerful Selective Kernel Tests for Feature Selection

Refining one's hypotheses in the light of data is a commonplace scientific practice, however, this approach introduces selection bias and can lead to specious statistical analysis. One approach of addressing this phenomena is via conditioning on the selection procedure, i.e., how we have used the data to generate our hypotheses, and prevents information to be used again after selection. Many selective inference (a.k.a. post-selection inference) algorithms typically take this approach but will "over-condition" for sake of tractability. While this practice obtains well calibrated $p$-values, it can incur a major loss in power. In our work, we extend two recent proposals for selecting features using the Maximum Mean Discrepancy and Hilbert Schmidt Independence Criterion to condition on the minimal conditioning event. We show how recent advances in multiscale bootstrap makes conditioning on the minimal selection event possible and demonstrate our proposal over a range of synthetic and real world experiments. Our results show that our proposed test is indeed more powerful in most scenarios.