Since standard candle bursts all have the same energy, the burst trigger is not biased towards more or less energetic bursts. Our method accounts for galaxies below the limiting magnitude, and further, host galaxies have been detected for all the optical transients observed thus far. Indeed, optical transients have been detected in only of the bursts localized by Beppo-SAX . A more systematic approach which uses all the observations to determine the redshift and burst energy distributions is warranted. Of course the probabilities used in the likelihoods must account for the detection thresholds.
Simulations show the resulting likelihoods are correct if the thresholds are understood.
For the forseeable future there are three cases. First, in the ideal case both the redshift and the fluence are known, and consequently the burst distance and its energy are determined directly. Sufficient bursts are required to constrain the distributions. The fluence distribution is the convolution of the redshift and energy distributions, therefore providing a powerful constraint on these distributions when analyzed in conjunction with bursts with more data. Finally, there are bursts for which the fluences and magnitudes of the apparent underlying host galaxies are measured.