authorea.com/65214

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}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 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}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).

Maximiliano Isiabout 1 year ago · Publicthese are two separate issue right? Also, Alan brought up the point that whether the prior dominates over the likelihood or not shouldn’t matter (you shouldn’t in principle just pick the prior that gives you the result you want a priori) and that we should justify this more strongly…