Controlled Discovery and Localization of Signals via Bayesian Linear Programming

Scientists often must simultaneously discover signals and localize them as precisely as possible. For instance, in genetic fine-mapping, high correlations between nearby genetic variants make it hard to identify the exact locations of causal variants. So the statistical task is to output as many disjoint regions containing a signal as possible, each as small as possible, while controlling false positives. Similar problems arise in any application where signals cannot be perfectly localized, such as locating stars in astronomical surveys and change point detection in time series data. Our first contribution is to propose a notion of resolution-adjusted power for such problems. Second, we introduce Bayesian Linear Programming (BLiP), a Bayesian method for jointly detecting and localizing signals. BLiP overcomes an extremely high-dimensional and non-convex problem to verifiably nearly maximize expected power while provably controlling false positives. BLiP is very computationally efficient and can wrap around nearly any Bayesian model and algorithm. Applying BLiP to existing state-of-the-art analyses of UK Biobank data (for genetic fine-mapping) and the Sloan Digital Sky Survey (for astronomical point source detection) increased resolution-adjusted power by 30-120% in just a few minutes of computation. BLiP is implemented in the new packages pyblip (Python) and blipr (R).