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Barium titanate, BaTiO₃ (BTO), has emerged as a promising electro-optic material with applications in silicon photonics. It boasts one of the largest known electro-optic coefficients; however, the origin of this giant electro-optic response has not been investigated in detail and is poorly understood. Here we report on a first principles study of the electro-optic or Pockels tensor in tetragonal P4mm BTO. We find good agreement with experiment if the P4mm structure is viewed as a dynamic average of four lower symmetry Cm structures. The large value of the Raman component of the EO coefficient is attributed to a low frequency and strong electron-phonon coupling of the lowest optical mode, and we trace the equally large piezoelectric contribution to the large components of the piezoelectric and elasto-optic tensors.