A new prior distribution for amplitudes


Currently in the known pulsar searches we have been using a uniform prior distribution between zero and some arbitrary, but large, upper value e.g.

\begin{equation} p(h_{0}|I)=\begin{cases}\frac{1}{h_{\mathrm{max}}}&\text{if }0\leq h_{0}<h_{\mathrm{max}}\\ 0&\text{otherwise}.\end{cases}\\ \end{equation}

However, this type of prior place a lot of probability at large values of \(h_{0}\) where in fact there is very little prior probability. As \(h_{0}\) is a scale factor, the general least informative prior is the Jeffreys prior of the form

\begin{equation} \label{eq:jeffreys} \label{eq:jeffreys}p(h_{0}|I)\propto\frac{1}{h_{0}},\\ \end{equation}

but this suffers from being improper (unnormalisable) and in the presence of no signal it will often overwhelm the likelihood when setting upper limits.

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 (inspired by Middleton et al., 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).