# A new prior distribution for amplitudes

## Introduction

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.

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

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

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

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.

$$p(h_{0}|\sigma,\mu,I)\propto\frac{1}{e^{(h_{0}-\mu)/\sigma}+1}.\\$$

To normalise this to get the probability density function (pdf) we see that

$$\int_{0}^{\infty}\frac{1}{e^{(h_{0}-\mu)/\sigma}+1}{\rm d}h_{0}=\sigma\log{\left(1+e^{\mu/\sigma}\right)},\\$$

so the actual pdf is

$$\label{eq:fermidirac} \label{eq:fermidirac}p(h_{0}|\sigma,\mu,I)=\frac{1}{\sigma\log{\left(1+e^{\mu/\sigma}\right)}}\left(e^{(h_{0}-\mu)/\sigma}+1\right)^{-1}.\\$$

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