  Understanding magnetic interactions in the parent compounds of
high-temperature superconductors forms the basis for determining their role for
the mechanism of superconductivity. For parent compounds of iron pnictide
superconductors such as $A$Fe$_2$As$_2$ ($A=$ Ba, Ca, Sr), although spin
excitations have been mapped out throughout the entire Brillouin zone (BZ),
measurements were carried out on twinned samples and did not allow for a
conclusive determination of the spin dynamics. Here we use inelastic neutron
scattering to completely map out spin excitations of $\sim$100\% detwinned
BaFe$_2$As$_2$. By comparing observed spectra with theoretical calculations, we
conclude that the spin excitations can be well described by an itinerant model
with important contributions from electronic correlations.
