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