  We use Bayesian methods to analyze the distribution of gamma ray burst
intensities reported in the Third BATSE Catalog (3B catalog) of gamma ray
bursts, presuming the distribution of burst sources (``bursters'') is
isotropic. We study both phenomenological and cosmological source distribution
models, using Bayes's theorem both to infer unknown parameters in the models,
and to compare rival models. We analyze the distribution of the time-averaged
peak photon number flux, F, measured on both 64 ms and 1024 ms time scales,
performing the analysis of data based on each time scale independently. Several
of our findings differ from those of previous analyses that modeled burst
detection less completely. In particular, we find that the width of the
intrinsic luminosity function for bursters is unconstrained, and the luminosity
function of the actually observed bursts can be extremely broad, in contrast to
the findings of all previous studies. Useful constraints probably require
observation of bursts significantly fainter than those visible to BATSE. We
also find that the 3B peak flux data do not usefully constrain the redshifts of
burst sources; useful constraints require the analysis of data beyond that in
the 3B catalog (such as burst time histories), or data from brighter bursts
than have been seen by BATSE (such as those observed by the Pioneer Venus
Orbiter). In addition, we find that an accurate understanding of the peak flux
distributions reported in the 3B almost certainly requires consideration of
data on the temporal and spectral properties of bursts beyond that reported in
the 3B catalog, and more sophisticated modeling than has so far been attempted.
