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.

\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.