We have investigated the electronic structure of the $p$-type diluted magnetic semiconductor In$_{1-x}$Mn$_x$As by photoemission spectroscopy. The Mn 3$d$ partial density of states is found to be basically similar to that of Ga$_{1-x}$Mn$_x$As. However, the impurity-band like states near the top of the valence band have not been observed by angle-resolved photoemission spectroscopy unlike Ga$_{1-x}$Mn$_x$As. This difference would explain the difference in transport, magnetic and optical properties of In$_{1-x}$Mn$_x$As and Ga$_{1-x}$Mn$_x$As. The different electronic structures are attributed to the weaker Mn 3$d$ - As 4$p$ hybridization in In$_{1-x}$Mn$_x$As than in Ga$_{1-x}$Mn$_x$As.