  We present all electron computations of the 3D Fermi surfaces (FS's) in
Ba$_{1-x}$K$_{x}$BiO$_{3}$ for a number of different compositions based on the
selfconsistent Korringa-Kohn-Rostoker coherent-potential-approximation
(KKR-CPA) approach for incorporating the effects of Ba/K substitution. By
assuming a simple cubic structure throughout the composition range, the
evolution of the nesting and other features of the FS of the underlying
pristine phase is correlated with the onset of various structural transitions
with K doping. A parameterized scheme for obtaining an accurate 3D map of the
FS in Ba$_{1-x}$K$_{x}$BiO$_{3}$ for an arbitrary doping level is developed. We
remark on the puzzling differences between the phase diagrams of
Ba$_{1-x}$K$_{x}$BiO$_{3}$ and BaPb$_{x}$Bi$_{1-x}$O$_{3}$ by comparing aspects
of their electronic structures and those of the end compounds BaBiO$_{3}$,
KBiO$_3$ and BaPbO$_3$. Our theoretically predicted FS's in the cubic phase are
relevant for analyzing high-resolution Compton scattering and
positron-annihilation experiments sensitive to the electron momentum density,
and are thus amenable to substantial experimental verification.
