  Using $\textit{ab-initio}$ crystal structure prediction we study the
high-pressure phase diagram of $\textit{A}BiO_3$ bismuthates ($A$=Ba, Sr, Ca)
in a pressure range up to 100$~$GPa. All compounds show a transition from the
low-pressure perovskite structure to highly distorted, low-symmetry phases at
high pressures (PD transition), and remain charge disproportionated and
insulating up to the highest pressure studied. The PD transition at high
pressures in bismuthates can be understood as a combined effect of steric
arguments and of the strong tendency of bismuth to charge-disproportionation.
In fact, distorted structures permit to achieve a very efficient atomic
packing, and at the same time, to have Bi-O bonds of different lengths. The
shift of the PD transition to higher pressures with increasing cation size
within the $\textit{A}BiO_3$ series can be explained in terms of chemical
pressure.
