To use this as a prior in the nested sampling code we need to be able to draw values uniformly from it. To do this we can use the cumulative probability distribution (cdf) of the function through inverse transform sampling.
The cdf, \(C(H_{0})\), is given by
\begin{align} C(H_{0}) & =\int_{0}^{H_{0}}p(h_{0}|\sigma,\mu,I){\rm d}h_{0}\notag \\ & =\int_{0}^{H_{0}}\frac{1}{\sigma\log{\left(1+e^{\mu/\sigma}\right)}}\left(e^{(h_{0}-\mu)/\sigma}+1\right)^{-1}{\rm d}h_{0}\notag \\ & =\frac{1}{\log{\left(1+e^{\mu/\sigma}\right)}}\left[\frac{H_{0}}{\sigma}+\log{\left(1+e^{-\mu/\sigma}\right)}-\log{\left(1+e^{(H_{0}-\mu)/\sigma}\right)}\right]\\ \end{align}A plot of this cdf can be seen in Figure \ref{fig:cdf}.