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In a high dimensional multiple testing framework, we present new confidencebounds on the false positives contained in subsets S of selected nullhypotheses. The coverage probability holds simultaneously over all subsets S,which means that the obtained confidence bounds are post hoc. Therefore, S canbe chosen arbitrarily, possibly by using the data set several times. We focusin this paper specifically on the case where the null hypotheses are spatiallystructured. Our method is based on recent advances in post hoc inference andparticularly on the general methodology of Blanchard et al. (2017); we buildconfidence bounds for some pre-specified forest-structured subsets {R k , k$\in$ K}, called the reference family, and then we deduce a bound for anysubset S by interpolation. The proposed bounds are shown to improvesubstantially previous ones when the signal is locally structured. Our findingsare supported both by theoretical results and numerical experiments. Moreover,we show that our bound can be obtained by a low-complexity algorithm, whichmakes our approach completely operational for a practical use. The proposedbounds are implemented in the open-source R package sansSouci.

Post hoc false positive control for spatially structured hypotheses