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Using $\textit{ab-initio}$ crystal structure prediction we study thehigh-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 thelow-pressure perovskite structure to highly distorted, low-symmetry phases athigh pressures (PD transition), and remain charge disproportionated andinsulating up to the highest pressure studied. The PD transition at highpressures in bismuthates can be understood as a combined effect of stericarguments and of the strong tendency of bismuth to charge-disproportionation.In fact, distorted structures permit to achieve a very efficient atomicpacking, and at the same time, to have Bi-O bonds of different lengths. Theshift of the PD transition to higher pressures with increasing cation sizewithin the $\textit{A}BiO_3$ series can be explained in terms of chemicalpressure.

$\textit{Ab-initio}$ study of $\textit{A}BiO_3$ ($A$=Ba, Sr, Ca) under high pressure