Scientists use high-dimensional measurement assays to detect and prioritize regions of strong signal in spatially organized domain. Examples include finding methylation enriched genomic regions using microarrays, and active cortical areas using brain-imaging. The most common procedure for detecting potential regions is to group neighboring sites where the signal passed a threshold. However, one needs to account for the selection bias induced by this procedure to avoid diminishing effects when generalizing to a population. This paper introduces pin-down inference, a model and an inference framework that permit population inference for these detected regions. Pin-down inference provides non-asymptotic point and confidence interval estimators for the mean effect in the region that account for local selection bias. Our estimators accommodate non-stationary covariances that are typical of these data, allowing researchers to better compare regions of different sizes and correlation structures. Inference is provided within a conditional one-parameter exponential family per region, with truncations that match the selection constraints. A secondary screening-and-adjustment step allows pruning the set of detected regions, while controlling the false-coverage rate over the reported regions. We apply the method to genomic regions with differing DNA-methylation rates across tissue. Our method provides superior power compared to other conditional and non-parametric approaches.
Keywords: Bump-hunting; Conditional inference; DNA-methylation; Non-stationary process; Selective inference; Spatial statistics.