Finite band inversion of ARPES in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ in comparison with optics
Using a maximum entropy technique within a finite band Eliashberg formalism we extract from recent high accuracy nodal direction angular resolved photo-emission spectroscopy (ARPES) data in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) a quasiparticle electron-boson spectral density. Both normal and superconducting state with d-wave gap symmetry are treated. Finite and infinite band results are considered and contrasted. We compare with results obtained for the related transport spectral density which follows from a similar inversion of optical data. We discuss the implication of our results for quasiparticle renormalizations in the antinodal direction.