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Bismuth iron garnet (BIG), i.e. Bi3Fe5O12, is a strong ferrimagnet that also possess outstanding magneto-optical properties such as the largest known Faraday rotation. These properties are related with the distribution of magnetic moments on octahedral and tetrahedral sites, the presence of spin gaps in the density of state and a strong spin-orbit coupling. In this work, first-principles ab initio calculations are performed to study the structural, electronic and magnetic properties of BIG using Density Functional Theory with Hubbard+U (DFT+U) correction including spin-orbit coupling and HSE06 hybrid functional. We found that the presence of spin gaps in the electronic structure results from the interplay between exchange and correlation effects and the crystal field strengths for tetrahedral and octahedral iron sublattices. The DFT+U treatment tends to close the spin-gaps for larger U due to over-localization effects, notably in the octahedral site. On the other hand, the hybrid functional confirms the occurrences of three spin gaps in the iron states of the conduction band as expected from optical measurements. A strong exchange splitting at the top of the valence bands associated with a lone-pair type mixture of O p and Bi s,p states is also obtained. Similar exchange splitting was not previously observed for other iron based garnets, such as for yttrium iron garnet. It follows that hole doping, as obtained by Ca substitution at Bi sites, results in a full spin polarized density at the Fermi energy. This work helps to shed more light on the theoretical comprehension of the properties of BIG and opens the route towards the use of advanced Many Body calculations to predict the magneto-optical coupling effects in BIG in a direct comparison with the experimental measurements.