  The main idea of nested sampling is to substitute the high-dimensional
likelihood integral over the parameter space $\Omega$ by an integral over the
unit line $[0,1]$ by employing a push-forward with respect to a suitable
transformation. For this substitution, it is often implicitly or explicitly
assumed that samples from the prior are uniformly distributed along this unit
line after having been mapped by this transformation. We show that this
assumption is wrong, especially in the case of a likelihood function with
plateaus. Nevertheless, we show that the substitution enacted by nested
sampling works because of more interesting reasons which we lay out. Although
this means that analytically, nested sampling can deal with plateaus in the
likelihood function, the actual performance of the algorithm suffers under such
a setting and the method fails to approximate the evidence, mean and variance
appropriately. We suggest a robust implementation of nested sampling by a
simple decomposition idea which demonstrably overcomes this issue.
