Inferring the spatial and energy distribution of gamma-ray burst sources. 1: Methodology
We describe Bayesian methods for analyzing the distribution of gamma-ray burst peak photon fluxes and directions. These methods fit the differential distribution, and have the following advantages over rival methods: (1) they do not destroy information by binning or averaging the data (as do, say, chi squared, the averaged value of V/V(sub max), and angular moment analyses); (2) they straightforwardly handle uncertainties in the measured quantities; (3) they analyze the strength and direction information jointly; (4) they use information available about nondetections; and (5) they automatically identify and account for biases and selection effects given a precise description of the experiment. In these methods, the most important information needed about the instrument threshold is not its value at the times of burst triggers, as is used in the average value of V/V(sub max) analyses, but rather the value of the threshold at times when no trigger occurred. We show that this information can be summarized as an average detection efficiency that is similar to the product of the exposure and efficiency reported in the First Burst and Transient Source Experiment (BATSE) Burst (1B) Catalog, but significantly different from it at low fluxes. We also quantify an important bias that results from estimating the peak flux by scanning the burst to find the peak number of counts in a window of specified duration, as was done for the 1B Catalog. When the duration of the peak of the light curve is longer than the window duration, a simple flux estimate based on the peak counts significantly overestimates the peak flux in a nonlinear fashion that distorts the shape of the log(N)-log(P) distribution. This distortion also corrupts analyses of the V/V(sub max) distribution that use ratios of counts above background to estimate V/V(sub max). The Bayesian calculation specifies how to account for this bias. Implementation of the Bayesian approach requires some changes in the way burst data are reported that we describe in detail. Subsequent papers will report analyses of the 1B Catalog data using the methods described here.