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

Recent gravitational-wave transient catalogs have used $p_\mathrm{astro}$, 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, $p_\mathrm{astro}$ 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 $p_\mathrm{astro}$, each employing a specific data reduction. While the range of $p_\mathrm{astro}$ 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 $\textit{unified } p_\mathrm{astro}$ 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 $p_\mathrm{astro}$ 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