Among the transparent conducting oxides, the perovskite barium stannate is most promising for various electronic applications due to its outstanding carrier mobility achieved at room temperature. Most important characteristics however, $i.e.$, its band gap, effective masses, and absorption edge remain controversial. Here, we provide a fully consistent picture by combining state-of-the-art $ab~initio$ methodology with forefront electron energy-loss spectroscopy (EELS) and optical absorption measurements. On- and off-axis valence EELS spectra, featuring signals originating from band gap transitions, are acquired on defect-free sample regions of a BaSnO$_{3}$ single crystal. These high-energy-resolution measurements are able to capture also very weak excitations below the optical gap, attributed to indirect transitions. By temperature-dependent optical absorption measurements, we assess band-gap renormalization effects induced by electron-phonon coupling. Overall, we find for the effective electronic mass, the direct and the indirect gap, the optical gap as well as the absorption onsets and spectra excellent agreement between both experimental techniques and the theoretical many-body results, supporting also the picture of a phonon-mediated mechanism where indirect transitions are activated by phonon-induced symmetry lowering. This work demonstrates a fruitful connection between different high-level theoretical and experimental methods for exploring the characteristics of advanced materials.
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