A better prior would be one that is flat(ish) over a range consistent with the detector data, but then falls off exponentially at higher values. Such a prior could have the form \citep[inspired by][]{Middleton_2015} of the Fermi-Dirac distribution, e.g.
\begin{equation} p(h_{0}|\sigma,\mu,I)\propto\frac{1}{e^{(h_{0}-\mu)/\sigma}+1}.\\ \end{equation}To normalise this to get the probability density function (pdf) we see that
\begin{equation} \int_{0}^{\infty}\frac{1}{e^{(h_{0}-\mu)/\sigma}+1}{\rm d}h_{0}=\sigma\log{\left(1+e^{\mu/\sigma}\right)},\\ \end{equation}so the actual pdf is
\begin{equation} \label{eq:fermidirac} \label{eq:fermidirac}p(h_{0}|\sigma,\mu,I)=\frac{1}{\sigma\log{\left(1+e^{\mu/\sigma}\right)}}\left(e^{(h_{0}-\mu)/\sigma}+1\right)^{-1}.\\ \end{equation}An example of this distribution, with \(\mu/\sigma=10\) and \(\sigma=1\) is shown in Figure \ref{fig:fermidirac}. It is worth noting that the point at which the distribution is half of its peak value (which always occurs at zero) is when \(h_{0}=\mu/\sigma\), which could be used to define the values to use in a particular case (e.g. being based on the standard deviation of the data).