INFERRING THE SPATIAL AND ENERGY-DISTRIBUTION OF GAMMA-RAY BURST SOURCES .1. METHODOLOGY

We describe Bayesian methods for analyzing the distribution of gamma-r ay burst peak photon fluxes and directions. These methods fit the diff erential distribution, and have the following advantages over rival me thods: (1) they do not destroy information by binning or averaging the data (as do, say, (2)(chi), [V/V-max], and angular moment analyses); (2) they straightforwardly handle uncertainties in the measured quanti ties; (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 gi ven a precise description of the experiment. In these methods the most important information needed about the instrument threshold is not it s value at the times of burst triggers, as is used in [V/V-max] analys es, but rather the value of the threshold at times when no trigger occ urred. We show that this information can be summarized as an average d etection efficiency that is similar to the product of the exposure and efficiency reported in the First BATSE Burst (1B) Catalog, but signif icantly 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 l ight curve is longer than the window duration, a simple nux estimate b ased on the peak counts significantly overestimates the peak flux in a nonlinear fashion that distorts the shape of the log (N)-log (P) dist ribution. This distortion also corrupts analyses of the V/V-max distri bution that use ratios of counts above background to estimate V/V-max. The Bayesian calculation specifies how to account for this bias. Impl ementation of the Bayesian approach requires some changes in the way b urst data are reported that we describe in detail. Subsequent papers w ill report analyses of the 1B Catalog data using the methods described here.