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Formation of intrinsic point defect in yttrium aluminum garnet (YAG) is comprehensively investigated using first-principles calculations. We showed that the defect formation energies of intrinsic point defects closely depend on the possible values of chemical potentials of Y, Al, and O. Both YAl16(a) anti-site defect, oxygen vacancy VO, and interstitial Oi might be the preferred defect species when chemical potentials of Y, Al, and O vary in different areas. The O-related defects are also important intrinsic point defects in YAG and deserve to be further examined. The oxygen self-diffusion are investigated and we found that energy barrier of oxygen diffusion is enhanced and decreased by neighboring YAl16(a) and AlY anti-site defects, respectively. The results are used to explain the mechanism of experimental observations that Al2O3 excess speeds up oxygen diffusion and that Y2O3 excess suppresses oxygen diffusion in YAG.