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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.

A method was developed for the detection and measurement of a periodic signal with unknown characteristics in a data set where the existence of such signal was not known. The method detects a signal by using Bayesian probability theory to compare a constant model for the signal to members of a class of models with periodic structure. The method was applied to simulated data generated with both stepwise and sinusoidal light curves, demonstrating that such signals can be sensitively detected and the signal frequency and its shape can be accurately estimated.

An exhaustive search of the Ginga data on gamma-ray burst GB870303 reveals two separate time intervals during which statistically significant line features are evident. One interval shows a single prominent line feature at about 20 keV; a second, shows two line features at about 20 and 40 keV. From model fits to the data, we find that both sets of lines are well-described by cyclotron resonant scattering in a magnetic field B about 1.8 x 10 exp 12 G, and that the differences in the line strengths between the two intervals are significant. The variations are qualitatively similar to those produced by a change in the viewing angle theta relative to the magnetic field. We conjecture that the change in theta is due to rotation of the neutron star, and derive limits P = 45-180 sec on the rotation period P.

Data on neutrinos from SN 1987A are compared here with parameterized models of the neutrino emission using a consistent and straightforward statistical methodology. The empirically measured detector background spectra are included in the analysis, and the data are compared with a much wider variety of neutrino emission models than was explored previously. It is shown that the inferred neutrino emission model parameters are strongly correlated. The analysis confirms that simple models of the neutrino cooling of the nascent neutron star formed by the SN adequately explain the data. The inferred radius and binding energy of the neutron star are in excellent agreement with model calculations based on a wide range of equations of state. The results also raise the upper limit of the electron antineutrino rest mass to roughly 25 eV at the 95 percent confidence level, roughly 1.5-5 times higher than found previously.

The paper describes an improved method for calculating gamma-ray spectra of gamma-ray bursts. This method is independent of any assumed model for the incident spectrum. In many cases it makes it possible to perform chi-squared model fitting without detailed knowledge of the instrument response functions. The results of extensive calculations using simulated data and realistic response functions are presented; the Backus-Gilbert technique produces model-independent spectral estimates that are quantitatively accurate and easy to calculate.

The implications of the detection of neutrinos from SN 1987A for the cooling of the nascent neutron star are considered. The nu-bar(e) number N, the apparent temperature, the cooling time scale measured by the Kamioka and IMB detectors, and the inferred neutron star apparent radius and binding energy are all found to provide striking verification of current supernova theory.