A Multilevel Empirical Bayesian Approach to Estimating the Unknown Redshifts of 1366 BATSE Catalog Long-Duration Gamma-Ray Bursts.

We present a catalog of the probabilistic redshift estimates and for 1366 individual Long-duration Gamma-Ray Bursts (LGRBs) detected by the Burst And Transient Source Experiment (BATSE). This result is based on a careful selection and modeling of the population distribution of 1366 BATSE LGRBs in the 5-dimensional space of redshift and the four intrinsic prompt gamma-ray emission properties: the isotropic 1024ms peak luminosity, the total isotropic emission, the spectral peak energy, as well as the intrinsic duration, while carefully taking into account the effects of sample incompleteness and the LGRB-detection mechanism of BATSE. Two fundamental plausible assumptions underlie our purely-probabilistic approach: 1. LGRBs trace, either exactly or closely, the Cosmic Star Formation Rate and 2. the joint 4-dimensional distribution of the aforementioned prompt gamma-ray emission properties is well-described by a multivariate log-normal distribution. Our modeling approach enables us to constrain the redshifts of individual BATSE LGRBs to within $0.36$ and $0.96$ average uncertainty ranges at $50\%$ and $90\%$ confidence levels, respectively. Our redshift predictions are completely at odds with the previous redshift estimates of BATSE LGRBs that were computed via the proposed phenomenological high-energy relations, specifically, the apparently-strong correlation of LGRBs' peak luminosity with the spectral peak energy, lightcurve variability, and the spectral lag. The observed discrepancies between our predictions and the previous works can be explained by the strong influence of detector threshold and sample-incompleteness in shaping these phenomenologically-proposed high-energy correlations in the literature.