Hierarchical Bayesian Inference on the intrinsic event rate of Type I gamma-ray bursts from the {\it Fermi}/GBM catalogue

The intrinsic event-rate of Type I gamma-ray bursts (GRBs) as a function of redshift, provides a probe for binary neutron star (BNS) mergers population. However, the above-mentioned inference is complicated by instrumental selection effects, the lack of redshift measurements for the majority of GRBs, and uncertainties in jet geometry. In this work, we applied a hierarchical Bayesian inference framework to the {\it Fermi}/GBM Type I GRB catalogue to reconstruct the underlying event-rate population. This framework accounts for the selection effects and marginalizes over the limited-information posteriors of individual GRB redshifts and jet structures. The population model uses parameterized luminosity function, distribution of jet structure parameters, the Madau-Dickinson star-formation rate and a delay-time distribution of BNS mergers. The posterior distribution of the hyperparameters in the model are to be inferred, using a GPU-accelerated MCMC algorithm. The intrinsic event rate density as function of redshift is constrained up to $z=5$, which a local rate density of $R_0=22.6_{-10.6}^{+15.0}\mathrm{G}\mathrm{p}{\mathrm{c}}^{-3}\mathrm{y}{\mathrm{r}}^{-1},$, which is consistent with the latest gravitational-wave measurements.