A Unified $p_\mathrm{astro}$ for Gravitational Waves: Consistently Combining Information from Multiple Search Pipelines

Recent gravitational-wave transient catalogs have used \pastro{}, the probability that a gravitational-wave candidate is astrophysical, to select interesting candidates for further analysis. Unlike false alarm rates, which exclusively capture the statistics of the instrumental noise triggers, \pastro{} incorporates the rate at which triggers are generated by both astrophysical signals and instrumental noise in estimating the probability that a candidate is astrophysical. Multiple search pipelines can independently calculate \pastro{}, each employing a specific data reduction. While the range of \pastro{} results can help indicate the range of uncertainties in its calculation, it complicates interpretation and subsequent analyses. We develop a statistical formalism to calculate a \emph{unified} \pastro{} for gravitational-wave candidates, consistently accounting for triggers from all pipelines, thereby incorporating extra information about a signal that is not available with any one single pipeline. We demonstrate the properties of this method using a toy model and by application to the publicly available list of gravitational-wave candidates from the first half of the third LIGO-Virgo-KAGRA observing run. Adopting a unified \pastro{} for future catalogs would provide a simple and easy-to-interpret selection criterion that incorporates a more complete understanding of the strengths of the different search pipelines