2 Methodology

We use the BATSE 4B GRB Catalog (Meegan et al. 1998), and BATSE bursts that occurred subsequent to the 4B catalog but before 1 May 1998. We also use the Ulysses supplement to the BATSE 4B catalog, which contains 219 BATSE bursts for which 3rd IPN annuli have been determined (Hurley et al. 1998). Hurley (private communication, 1998) has kindly made available at our request 3rd IPN annuli for an additional 9 BATSE bursts that occurred subsequent to the period of the BATSE 4B catalog but before 1 May 1998.

We have compiled three Type I SNe samples. The first is a sample of 37 Type Ia SNe at low redshift (z < 0.1) from the CfA SN Search Team (Riess 1998, private communication). The second is a sample of 46 moderate redshift (0.1 < z < 0.830) Type Ia SNe from the Supernova Cosmology Project (Perlmutter 1998, private communication). The third sample consists of 20 Type Ib, Ib/c, and Ic SNe compiled from IAU Circulars and various SNe catalogs.

We compare three hypotheses: H₁: The association between SNe and GRBs is real. If a SN is observed, there is a chance $\epsilon$ that BATSE sees the associated GRB, where $\epsilon$ is the average BATSE temporal exposure. While $\epsilon$ varies with Declination, the variation is modest and we neglect it. The probability density for the time of occurrence of the ith supernova is assumed uniform in the interval $[t_i,t_i+\tau_i]$, so that all GRBs that occur in that interval have an equal prior probability of being associated with the SN. $H_1^\prime$: The association is real, but only a fraction f of detectable SNe produce detectable GRB. H₂: There is no association between SNe and GRBs.

We calculate the Bayesian Odds, $\ensuremath{\mathcal{O}}$ favoring H₁ over H₂. The details of the calculation are presented in (Graziani et al. 1998). We separately compare $H_1^\prime$ to H₂, denoting the odds favoring $H_1^\prime$ over H₂ by $\ensuremath{\mathcal{O}}^\prime$.Finally, we calculate the posterior probability distribution for the parameter f in the model $H_1^\prime$, and infer an upper limit on its value.