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Transparent oxides are essential building blocks to many technologies, ranging from components in transparent electronics, transparent conductors, to absorbers and protection layers in photovoltaics and photoelectrochemical devices. However, thus far, it has been difficult to develop p-type oxides with wide band gap and high hole mobility; current state-of-art transparent p-type oxides have hole mobility in the range of < 10 cm$^2$/Vs, much lower than their n-type counterparts. Using high-throughput computational screening to guide the discovery of novel oxides with wide band gap and high hole mobility, we report the computational identification and the experimental verification of a bismuth-based double-perovskite oxide that meets these requirements. Our identified candidate, Ba$_2$BiTaO$_6$, has an optical band gap larger than 4 eV and a Hall hole mobility above 30 cm$^2$/Vs. We rationalize this finding with molecular orbital intuitions; Bi$^{3+}$ with filled s-orbitals strongly overlap with the oxygen p, increasing the extent of the metal-oxygen covalency and effectively reducing the valence effective mass, while Ta$^{5+}$ forms a conduction band with low electronegativity, leading to a high band gap beyond the visible range. Our concerted theory-experiment effort points to the growing utility of a data-driven materials discovery and the combination of both informatics and chemical intuitions as a way to discover future technological materials.