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\section{A new prior}  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][]{2015arXiv150700992M} by][]{Middleton_2015}  of the \href{https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac\_statistics}{Fermi-Dirac distribution}, e.g. \begin{equation}  p(h_0|\sigma, \mu, I) \propto \frac{1}{e^{(h_0-\mu)/\sigma} + 1}.  \end{equation}