Photoelectron spectroscopy is a technique to obtain the binding energy of the electrons inside material.
It is "photon in"- "electron out" technique. For the detail of the Photoelectron spectroscopy, refer to the ARPES page.
In the case of inelastic photoelectron spectroscopy, the energy conservation rule
hν = EB +Ekin + Φ
is violated due to the interaction between photoelectrons and phonons or between photoholes and phonons.
By measuring the energy of the inelastic photoelectrons, we can see the details of these interactions