Geometric Approach to Analytic Marginalisation of the Likelihood Ratio for Continuous Gravitational Wave Searches

Abstract

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, and the other representing the maximised likelihood ratio or F-statistic. Analytic marginalisation over the angle between the vectors yields a marginalised likelihood ratio, which is a function of the 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 had identical detection power to the F-statistic. This approach demonstrates a route to viewing the F-statistic in a Bayesian context, while retaining the advantages of its efficient computation.

Improved short-segment detection statistic for continuous gravitational waves

“…An alternative derivation of the F-statistic as a Bayes factor [11] reveals the underlying amplitude priors to be unphysical, which is why the F-statistic is not statistically optimal. The main advantage of the F-statistic is the analytical elimination of the four amplitude parameters, which gives it a computational advantage over any alternative that would require explicit numerical operations to deal with the unknown amplitude parameters (e.g., see [12][13][14] for further discussion).…”

Search Methods for Continuous Gravitational-Wave Signals from Unknown Sources in the Advanced-Detector Era

“…The F -statistic is a standard detection statistic for CW signals. Initially derived as a maximum-likelihood estimator with respect to amplitude parameters [21,70], it was later rederived in a Bayesian context as a Bayes factor, gauging the presence (or lack) of a signal in a Gaussian noise data stream, in which amplitude parameters are marginalized using a rather unphysical set of amplitude priors [71][72][73][74][75][76]. This detection statistic can be extended for more generic types of sources, such as binary white-dwarf systems [77] or the inspiral phase of binary black-hole coalescences [78].…”

Search methods for continuous gravitational-wave signals from unknown sources in the advanced-detector era

“…The F -statistic is a standard detection statistic for CW signals. Initially derived as a maximum-likelihood estimator with respect to amplitude parameters [21,70], it was later rederived in a Bayesian context as a Bayes factor, gauging the presence (or lack) of a signal in a Gaussian noise data stream, in which amplitude parameters are marginalized using a rather unphysical set of amplitude priors [71][72][73][74][75][76]. This detection statistic can be extended for more generic types of sources, such as binary white-dwarf systems [77] or the inspiral phase of binary black-hole coalescenses [78].…”