  We report an Fe $K$-edge resonant inelastic X-ray scattering (RIXS) study of
K$_{0.83}$Fe$_{1.53}$Se$_2$. This material is an insulator, unlike many parent
compounds of iron-based superconductors. We found a sharp excitation around 1
eV, which is resonantly enhanced when the incident photon energy is tuned near
the pre-edge region of the absorption spectrum. The spectral weight and line
shape of this excitation exhibit clear momentum dependence. In addition, we
observe momentum-independent broad interband transitions at higher excitation
energy of 3-7 eV. Calculations based on a 70 band $dp$ orbital model, using a
moderate $U_{\rm eff}\approx 2.5$ eV, indicate that the $\sim$1 eV feature
originates from the correlated Fe 3$d$ electrons, with a dominant $d_{xz}$ and
$d_{yz}$ orbital character. We find that a moderate $U_{\rm eff}$ yields a
satisfying agreement with the experimental spectra, suggesting that the
electron correlations in the insulating and metallic iron based superconductors
are comparable.
