  We investigate the physical properties of the Ba$_{1-x}$K$_x$BiO$_3$
compounds with a focus on the optical properties. Results from the simple
Holstein model, describing a single band coupled to an oxygen breathing mode
with parameters derived from first principles calculations, are in excellent
agreement with a broad range of experimental information. It accounts for an
insulating parent compound at $x=0$ %, with a direct- (optical) and an
indirect-gap, and a metal insulator transition around $x = 0.38$. Strong
electron-phonon coupling leads to spectral weight redistribution over a
frequency scale much larger than the characteristic phonon frequency and to
strongly anharmonic phonons. We find that the metallic phase in the vicinity of
phase boundary is close to the polaronic regime.
