  The likelihood ratio for a continuous gravitational wave signal is viewed
geometrically as a function of the orientation of two vectors; one representing
the optimal signal-to-noise ratio, the other representing the maximised
likelihood ratio or $\mathcal{F}$-statistic. Analytic marginalisation over the
angle between the vectors yields a marginalised likelihood ratio which is a
function of the $\mathcal{F}$-statistic. Further analytic marginalisation over
the optimal signal-to-noise ratio is explored using different choices of prior.
Monte-Carlo simulations show that the marginalised likelihood ratios have
identical detection power to the $\mathcal{F}$-statistic. This approach
demonstrates a route to viewing the $\mathcal{F}$-statistic in a Bayesian
context, while retaining the advantages of its efficient computation.
