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.

Structural equation models and Bayesian networks have been widely used to study causal relationships between continuous variables. Recently, a non-Gaussian method called LiNGAM was proposed to discover such causal models and has been extended in various directions. An important problem with LiNGAM is that the results are affected by the random sampling of the data as with any statistical method. Thus, some analysis of the statistical reliability or confidence level should be conducted. A common method to evaluate a confidence level is a bootstrap method. However, a confidence level computed by ordinary bootstrap method is known to be biased as a probability-value ($p$-value) of hypothesis testing. In this paper, we propose a new procedure to apply an advanced bootstrap method called multiscale bootstrap to compute confidence levels, i.e., p-values, of LiNGAM outputs. The multiscale bootstrap method gives unbiased $p$-values with asymptotic much higher accuracy. Experiments on artificial data demonstrate the utility of our approach.