Here, I will describe an alternative notion of coverage that focuses on capturing the smooth structure of a function. I will develop procedures that yield adaptive confidence bands, where coverage is understood in this alternative sense, and show how such bands are used in practice. I will illustrate the approach and surrounding issues with an analysis of supernova data to make inferences about dark energy and the acceleration of the universe.
The LAT is a pair-conversion instrument. It consists of a tracker - a four-by-four array of towers, where each tower consists of alternating conversion and detection layers. The conversion layers are tungsten foils, and the detection layers consist of silicon microstrips, with separate planes measuring x- and y- position. Beneath the towers is a segmented calorimeter. In this work we focus on the tracker, which is designed to estimate the direction of incident photons.
Incident gamma-rays convert to electron-positron pairs in the tungsten foils, and these charged particles trigger the silicon microstrips as they traverse the remaining layers. They also produce secondary charged particles, and secondary photons (which may undergo pair conversion later). The charged particles are scattered as they traverse each layer, losing energy and changing direction. To reconstruct the direction of the incident photon the energy split between the initial electron and positron must be well estimated.
The statistics of each simple interaction are well quantified. We describe work-in-progress on developing a fully statistical reconstruction methodology, that incorporates in detail the statistics of the simple interactions to compute the full pdf over the energy and direction of the incident photons. It uses model selection methods to estimate the probabilities of the possible geometrical configurations of the particles produced in the detector, and numerical marginalization over the energy loss and scattering angles at each layer.