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

Sharan Banagiri<EMAIL_ADDRESS>Center for Interdisciplinary
Exploration and Research in Astrophysics, Northwestern University, 1800
Sherman Avenue, Evanston, Illinois 60201, USA  Christopher P. L. Berry
SUPennsylvania, School of Physics and Astronomy, University of Glasgow,
Glasgow G12 8QQ, United Kingdom  Gareth S. Cabourn Davies  University of
Portsmouth, Portsmouth PO1 3FX, United Kingdom  Leo Tsukada  Department of
Physics, The Pennsylvania State University, University Park, Pennsylvania
16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, Pennsylvania 16802, USA  Zoheyr Doctor  Center
for Interdisciplinary Exploration and Research in Astrophysics, Northwestern
University, 1800 Sherman Avenue, Evanston, Illinois 60201, USA

###### Abstract

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
_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

##  I Introduction

The detection of gravitational waves Abbott  _et al._ (2016a, 2021a) by the
Laser Interferometer Gravitational-Wave Observatory (LIGO) Aasi  _et al._
(2015) and Virgo Acernese  _et al._ (2015) detectors is the culmination of
decades of research. Not only are sensitive detectors needed to the measure
the GW signals Pitkin  _et al._ (2011); Abbott  _et al._ (2020a), but
sophisticated data analysis is needed to distinguish astrophysical signals
from detector noise Abbott  _et al._ (2020b). In the case of transient
signals, such as those from compact binary coalescences, detection algorithms
identify candidate signals by matching template signals to the data Abbott
_et al._ (2016b) or by looking for coherent signals in multiple detectors
Abbott  _et al._ (2016c). Only after identifying GW candidates in detector
data can we start to understand the astrophysical population of GW sources.

A fundamental question in any data analysis problem is the veracity of the
signal or the effect it is considering. The statistical significance of a
transient GW is generally assessed by calculating how likely the data would
appear due to noise fluctuations. This may be quantified by the false alarm
rate (FAR) or by a p value. Statistics like FAR are particularly well suited
for making a first detection, where we do not know the population of signals.
However, a more complete assessment of the probability that a candidate is
real can be obtained by considering not only the FAR but also the _true_ alarm
rate (i.e., how often the algorithm would find identify a real signal) whose
calculation requires knowledge of the source population and our sensitivity to
their signals, and hence is subject to additional uncertainties not inherent
in the calculation of the FAR or p value. By combining the false and true
alarm rates, we can calculate the probability of astrophysical origin
$p_{\mathrm{astro}}$ for a candidate Farr  _et al._ (2015); Guglielmetti  _et
al._ (2009). In the GW literature, $p_{\mathrm{astro}}$ was first used to
estimate the astrophysical probability of GW150914 and the then-considered
marginal trigger LVT151012 Abbott  _et al._ (2016d, e) (now known as GW151012
Abbott  _et al._ (2019a)). The probability of astrophysical origin
$p_{\mathrm{astro}}$ directly addresses the key question of how probable a
candidate is to be real accounting for both our understanding of detector
sensitivity and the population of GW sources.

We are now in the era of having many GW candidates, such that we are building
an understanding of the source population Abbott  _et al._ (2021b). GWs
catalogs from both the LIGO-Virgo-KAGRA (LVK) Collaboration Abbott  _et al._
(2019a, 2021a) and independent teams Nitz  _et al._ (2021a); Olsen  _et al._
(2022) now commonly use $p_{\mathrm{astro}}$ as a criterion to identify
interesting candidates for further analysis. By the end of their third
observing run (O3), the LVK Collaboration have reported $90$ CBC candidates
with $p_{\mathrm{astro}}{}{}>0.5$ Abbott  _et al._ (2019a, 2021c, 2021d,
2021a). Despite the additional uncertainties inherent in its calculation,
$p_{\mathrm{astro}}$ is important to consider now that we now have
observations from multiple observing runs of different sensitivity Abbott  _et
al._ (2020a) and observations of different types of sources Abbott  _et al._
(2016a, 2017a, 2021e, 2021a). At a given FAR, a candidate from the most recent
O3 run Buikema  _et al._ (2020); Acernese  _et al._ (2022) is more probable to
be real than a candidate from the (less sensitive) first observing run Abbott
_et al._ (2016f). Similarly, at a given FAR, a candidate consistent with
originating from a $30M_{\odot}+30M_{\odot}$ binary black hole (BBH) merger is
more probable to be real than a candidate consistent with a binary neutron
star (BNS) origin Kapadia  _et al._ (2020) because the former can be detected
from a greater distance and so are more frequently observed Abbott  _et al._
(2021a). Since $p_{\mathrm{astro}}$ accounts for the true alarm rate, it can
naturally be used when compiling a heterogeneous catalog of diverse sources
from different observing runs.

During O3, the LVK Collaboration used four search pipelines for its final
analysis Abbott  _et al._ (2021a), with each calculating its own
$p_{\mathrm{astro}}$: there are three CBC matched-filtered pipelines (GstLAL
Messick  _et al._ (2017); Sachdev  _et al._ (2019); Hanna  _et al._ (2020);
Cannon  _et al._ (2021); Kapadia  _et al._ (2020), PyCBC Allen  _et al._
(2012); Dal Canton  _et al._ (2014); Usman  _et al._ (2016); Nitz  _et al._
(2017); Davies  _et al._ (2020); Dent  and MBTA Adams  _et al._ (2016); Aubin
_et al._ (2021); Andres  _et al._ (2022)), and non-template-based search
pipeline (cWB Klimenko and Mitselmakher (2004); Klimenko  _et al._ (2011,
2016)) that makes minimal assumptions about the transient signals. Multiple
analyses with different choices help to explore the GW-signal space more
thoroughly, making it easier to find a diverse range of signals.
Unfortunately, this complicates the use of $p_{\mathrm{astro}}$ when
interpreting current GW candidates in three ways. First, there are differences
in the assumptions about the source population between analyses that mean that
results are not directly comparable. Second, as illustrated in Fig. 6 of
GWTC-3 Abbott  _et al._ (2021a), pipelines can have different relative
sensitivities, which may provide valuable insights into whether the candidate
is real. This information is not incorporated in individual
$p_{\mathrm{astro}}$ estimates. Third, it means there is no single assessment
of the significance of a candidate, complicating the calculation of
contamination and assessment of search sensitivity (and hence selection
effects, which are essential for population studies). Indeed, as can be seen
in Table 1 of GWTC-3 Abbott  _et al._ (2021a), pipelines can often find a
different set of triggers over the same stretch of data and can sometimes
calculate significantly different $p_{\mathrm{astro}}$ values for the same
candidate. To fully make use of the different search pipelines, it is
necessary to combine their results to produce a single $p_{\mathrm{astro}}$
value for each candidate.

Some early work in combining results from multiple pipelines was done in the
pre-detection era. For example, developing a way to estimate generalized
frequentist upper limits with multiple pipelines Sutton (2009), and developing
unified detection statistics Biswas  _et al._ (2012). In this paper, we
present a framework that can combine the results from multiple detection
algorithms to produce a unified $p_{\mathrm{astro}}$ employing information
about the correlation between pipelines. This means that catalogs can be
produced consistently using a simple threshold on this statistic and that
downstream users may more straightforwardly calculate the contamination
fraction for such a catalog. This statistic is amenable to use in population
inferences that incorporate low-significance candidates Galaudage  _et al._
(2020); Roulet  _et al._ (2020). Crucially, as the formalism accounts for the
correlations between different pipeline outputs (or lack thereof), we can make
use of the full ensemble of results when evaluating whether a candidate is
real. We present an illustration of the properties of our $p_{\mathrm{astro}}$
formalism, and provide a proof-of-concept example of its use with real GW data
from the first half of O3 (henceforth called O3a) analyzed for GWTC-2.1 Abbott
_et al._ (2021d).

In Sec. II we define $p_{\mathrm{astro}}$ from first principles and describe
its connection with the Poisson mixture-model formalism, first developed by
Farr et al. Farr  _et al._ (2015) (henceforth called the FGMC method) in the
GW context. Section III extends this formalism to calculate unified
$p_{\mathrm{astro}}$ from multiple search pipelines. We demonstrate some of
its useful properties by the means of a simple toy model in Sec. IV, followed
by an illustrative application to triggers from GWTC-2.1 in Sec. V. This
illustration highlights the tools that need to be developed in order to
produce a reliable unified $p_{\mathrm{astro}}$ for use in future GW catalogs.
We discuss the applications and extensions of our formalism in Sec. VI.

##  II Defining $p_{\mathrm{astro}}$

In this section, we present a pedagogical derivation for $p_{\mathrm{astro}}$
as commonly seen in literature Farr  _et al._ (2015); Kapadia  _et al._
(2020); Abbott  _et al._ (2016d, e), connecting it the Poisson mixture-model
FGMC formalism. In Sec. III, we explain how to extend this for the case of
multiple search pipelines.

The primary goal of a search pipeline is to identify candidates of
astrophysical origin. Pipelines usually accomplish this by maximizing some
statistics over a small stretch of data, and identifying a particular
candidate, which we will call a trigger. We assume that the stretch of data
contains at most one signal. Every trigger has a detection statistic $x$
associated with it by the search pipeline.

The distribution of the detection statistic under pure noise lets us calculate
the FAR distribution of the pipeline. In the context of GW analysis, this can
be done by the bootstrapping method of time slides, where the data stream from
one detector is shifted with respect to another by more than the light-travel
time between them Babak  _et al._ (2013), or by constructing a model of the
noise using triggers that are not coincident between detectors Cannon  _et
al._ (2013).

We define $p_{\mathrm{astro}}$ as the probability that a particular trigger is
caused by an astrophysical signal, as opposed to a noise fluctuation or
terrestrial contamination. Hence $p_{\mathrm{astro}}$ is properly associated
with the trigger and its detection statistic (one specific reduction of the
data) rather than directly the underlying data itself. When different
pipelines analyze the data, they make different analysis choices which give
them differing sensitivities, and so it is possible that they yield different
triggers and $p_{\mathrm{astro}}$ values.

Since a trigger can be caused by an astrophysical signal or by noise,
$p_{\mathrm{astro}}$ depends on the posterior probability for these two
hypotheses. We define $p(\mathcal{S}|x,\Phi_{\mathrm{s}},\Phi_{\mathrm{n}})$
as the posterior probability for the signal hypothesis $\mathcal{S}$, and
$p(\varnothing|x,\Phi_{\mathrm{n}})$ as the posterior probability for the
noise hypothesis $\varnothing$. Both these probabilities are conditioned on
the detection statistic of the trigger $x$, and assume some signal and noise
parameters, $\Phi_{\mathrm{s}}$ and $\Phi_{\mathrm{n}}$. We can then write
$p_{\mathrm{astro}}$ for the trigger as the normalized probability that it is
astrophysical,

$p_{\mathrm{astro}}{}(x)=\frac{p(\mathcal{S}|x,\Phi_{\mathrm{s}},\Phi_{\mathrm{n}})}{p(\mathcal{S}|x,\Phi_{\mathrm{s}},\Phi_{\mathrm{n}})+p(\varnothing|x,\Phi_{\mathrm{n}})}.$
(1)

We shall suppress the signal and noise parameters from here on for the purpose
of clarity. A value $p_{\mathrm{astro}}{}{}=1$ implies perfect confidence
about the presence of a signal in an underlying segment of data, while
$p_{\mathrm{astro}}{}{}=0$ implies perfect confidence of its absence, and
$p_{\mathrm{astro}}{}{}=0.5$ implies no preference between the signal and
noise hypothesis.

The posterior probabilities for two cases are difficult to compute directly.
What is more easily accessible is the likelihood of the trigger $x$, under the
signal and noise hypotheses. Therefore using Bayes’ theorem, we write
$p(\mathcal{S}|x)$ and $p(\varnothing|x)$ as

$\begin{split}p(\mathcal{S}|x)=\frac{\pi_{\mathrm{s}}p(x|\mathcal{S})}{Z(d)},\\\
p(\varnothing|x)=\frac{\pi_{\mathrm{n}}p(x|\varnothing)}{Z(d)},\\\
\end{split}$ (2)

where $p(x|\mathcal{S})$ and $p(x|\varnothing)$ are the likelihoods for
getting a trigger with statistic $x$ under the signal and noise hypothesis,
respectively; $\pi_{\mathrm{s}}$ and $\pi_{\mathrm{n}}$ are the corresponding
priors, and $Z(d)$ is the Bayesian evidence. We can then write
$p_{\mathrm{astro}}$ as

$p_{\rm
astro}(x)=\frac{\pi_{\mathrm{s}}p(x|\mathcal{S})}{\pi_{\mathrm{s}}p(x|\mathcal{S})+\pi_{\mathrm{n}}p(x|\varnothing)}.$
(3)

Suppose that we analyze a long stretch of data with a single search pipeline.
Let $R_{\mathrm{s}}$ and $R_{\mathrm{n}}$ be the rates of astrophysical and
noise-only triggers, respectively, assuming a given detector sensitivity,
operating characteristics, and algorithmic choice (including any preliminary
thresholds that are applied) made by the search pipeline. We define the signal
and noise count parameters, $\Lambda_{\mathrm{s}}{}{}$ and
$\Lambda_{\mathrm{n}}{}{}$, respectively. These are the mean number of
astrophysical and noise triggers (above the predefined thresholds) for an
observing duration $T$ (not necessarily the number of triggers in any
particular realization of the data) such that

$\Lambda_{\mathrm{s}}{}{}=R_{\mathrm{s}}T\qquad\text{ and
}\qquad\Lambda_{\mathrm{n}}{}{}=R_{\mathrm{n}}T.$ (4)

The total number of triggers ${N}$ will follow a Poisson distribution with a
count parameter $\Lambda=\Lambda_{\mathrm{s}}{}{}+\Lambda_{\mathrm{n}}{}{}$:

$p({N}|\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})=\frac{\left(\Lambda_{\mathrm{s}}{}{}+\Lambda_{\mathrm{n}}{}{}\right)^{N}\,e^{-\left(\Lambda_{\mathrm{s}}{}{}+\Lambda_{\mathrm{n}}{}{}\right)}}{{N}!}.$
(5)

For computational reasons, it is common to require that the trigger has to
pass some primary statistical threshold, before being used for further
analysis. For example, the GWTC-2.1 Abbott  _et al._ (2021d) and GWTC-3 Abbott
_et al._ (2021a) analyses considered only triggers that had a FAR $\leq
2~{}\mathrm{day}^{-1}$. Our formalism does not make any strong assumptions
about the preliminary threshold, as long as it is consistently used among all
pipelines. Thresholds are incorporated by suitably defining the counts
$\Lambda_{\mathrm{s}}{}{}$ and $\Lambda_{\mathrm{s}}{}{}$ as for triggers that
pass the threshold.

The count parameters are generally assumed to be unknown and have to be
measured empirically using the triggers. To do this we use the Poisson
mixture-model FGMC formalism:

$p(\\{x\\}|\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{},N)=\prod_{i}^{N}\left[\pi_{\mathrm{s}}p({x_{i}}|\mathcal{S})+\pi_{\mathrm{n}}p({x_{i}}|\varnothing)\right],$
(6)

where $\\{x\\}$ denotes the set of all triggers found in the data. Conditioned
upon $\Lambda_{\mathrm{s}}{}{}$ and $\Lambda_{\mathrm{n}}{}{}$, the prior
probabilities are just the relative rate of the triggers of that category,

$\pi_{\mathrm{s}}=\frac{\Lambda_{\mathrm{s}}{}{}}{\Lambda_{\mathrm{s}}{}{}+\Lambda_{\mathrm{n}}{}{}},\qquad\pi_{\mathrm{n}}=\frac{\Lambda_{\mathrm{n}}{}{}}{\Lambda_{\mathrm{s}}{}{}+\Lambda_{\mathrm{n}}{}{}}.$
(7)

Therefore, Eq. (6) becomes

$p(\\{x\\}|\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{},N)=\frac{\prod_{i}^{N}\left[\Lambda_{\mathrm{s}}{}{}p({x_{i}}|\mathcal{S})+\Lambda_{\mathrm{n}}{}{}p({x_{i}}|\varnothing)\right]}{(\Lambda_{\mathrm{s}}{}{}+\Lambda_{\mathrm{n}}{}{})^{{N}}}.$
(8)

Using Bayes’ theorem, we can relate the posterior distribution on the count
parameters with the the Poisson likelihood,

$\begin{split}p(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{}|\\{x\\},{N})&\propto
p(\\{x\\},{N}|\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})\,\pi(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})\\\
&\propto
p(\\{x\\}|{N},\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})\,p({N}|\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})\pi(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{}),\end{split}$
(9)

where $\pi(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})$ is the prior on
the mean count. Putting this together with Eq. (5), we obtain the posterior
distributions for $\Lambda_{\mathrm{s}}{}{}$ and $\Lambda_{\mathrm{n}}{}{}$,

$\begin{split}p(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{}|\\{x\\},{N})\propto\,&e^{-(\Lambda_{\mathrm{n}}{}{}+\Lambda_{\mathrm{s}}{}{})}\pi(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})\times\\\
&\prod_{i}^{N}\left\\{\Lambda_{\mathrm{s}}{}{}p({x_{i}}|\mathcal{S})+\Lambda_{\mathrm{n}}{}{}p({x_{i}}|\varnothing)\right\\}.\end{split}$
(10)

Finally, combining Eq. (3) and Eq. (7), we can calculate a
$p_{\mathrm{astro}}$ that marginalizes over the posterior of
$\Lambda_{\mathrm{s}}{}{}$ and $\Lambda_{\mathrm{n}}{}{}$,

$p_{\rm
astro}(x)=\int\mathrm{d}\Lambda_{\mathrm{s}}{}{}\mathrm{d}\Lambda_{\mathrm{n}}{}{}\frac{\Lambda_{\mathrm{s}}{}{}p(x|\mathcal{S})\,p(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{}|\\{x\\},N)}{\Lambda_{\mathrm{s}}{}{}p(x|\mathcal{S})+\Lambda_{\mathrm{n}}{}{}p(x|\varnothing)}.$
(11)

This fundamental framework has been used by all search pipelines both by the
LVK Collaboration and other groups in compiling GW catalogs Abbott  _et al._
(2016d, e, 2021a, 2021d); Olsen  _et al._ (2022); Venumadhav  _et al._ (2020);
Nitz  _et al._ (2020, 2021b, 2021a).

##  III Towards a unified $p_{\mathrm{astro}}$

We now develop a way to combine information from multiple pipelines for
calculating a unified $p_{\mathrm{astro}}$. Suppose that we have several
search pipelines that yield triggers after running over the same underlying
data. Consider any two corresponding triggers $x^{\alpha}$ and $x^{\beta}$
from pipelines $\alpha$ and $\beta$; as these triggers are produced by
different pipelines they may be associated with different statistics, but
these statistics will be correlated depending upon the relative sensitivities
of the two pipelines. Understanding the correlations between pipelines is key
to understanding how to construct a unified $p_{\mathrm{astro}}$.

The correlations between different search pipelines are not typically
accounted for when compiling GW results. In some analyses, the FAR is
multiplied by a trials factor equal to the number of different searches Abbott
_et al._ (2017b, 2022). However, this is a conservative choice: it would be
correct if the results were noise triggers that were uncorrelated, but in
general, we would expect some correlation since search pipelines are searching
for similar signals in the data. This correlation should reduce the effective
trials factor (in the limit of running two identical pipelines there would be
no need to add a trials factor). Our framework accounts for correlations in
both noise triggers and signal triggers to construct a unified
$p_{\mathrm{astro}}$.

Let us define $\vec{x}=\\{x^{\alpha},x^{\beta},....\\}$ as the triggers that
correspond with each other from a series of different pipelines. For the rest
of this paper, a vector usually means a vector of pipeline outputs, such that
$\\{\vec{x}\\}$ is the set of all triggers from all the pipelines in the
stretch of data being analyzed, and a latin index is used to indicate the
individual data segment or trigger. As in Eq. (2), the probability
$p(\mathcal{S}|\vec{x})$ is given by,

$p(\mathcal{S}|\vec{x})\propto\Lambda_{\mathrm{s}}{}{}p(\vec{x}|\mathcal{S}).$
(12)

Similarly, for the noise hypothesis,

$p(\varnothing|\vec{x})\propto\Lambda_{\mathrm{n}}{}{}p(\vec{x}|\varnothing).$
(13)

Here, $p(\vec{x}|\mathcal{S})$ and $p(\vec{x}|\varnothing)$ are joint
likelihoods for obtaining $\vec{x}=\\{x^{\alpha},x^{\beta},....\\}$ and are
dependent on the aforementioned correlations between trigger statistics across
pipelines. If we learn these joint likelihood distributions, using simulated
signals and noise triggers, we can calculate a unified $p_{\mathrm{astro}}$.

While several methods to learn arbitrary distributions exist, we will
primarily use kernel density estimation (KDE) methods from scikit-learn
Pedregosa  _et al._ (2011) to learn $p(\vec{x}|\mathcal{S})$ and
$p(\vec{x}|\varnothing)$. The choice of method used to reconstruct the
distribution is important, and the distributions must be properly
characterized to successfully calculate a reliable $p_{\mathrm{astro}}$.
However, for the purposes of illustrating the framework needed to compute a
unified $p_{\mathrm{astro}}$, this method may be treated as a black box.

Once the distribution of these correlations is learned, one can modify Eq.
(10) for a joint FGMC estimate

$\begin{split}p(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{}|\\{\vec{x}\\},{N})\propto\,&e^{-(\Lambda_{\mathrm{n}}{}{}+\Lambda_{\mathrm{s}}{}{})}\pi(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{})\times\\\
&\prod_{i}^{N}\left\\{\Lambda_{\mathrm{s}}{}{}p({\vec{x}_{i}}|\mathcal{S})+\Lambda_{\mathrm{n}}{}{}p({\vec{x}_{i}}|\varnothing)\right\\}.\end{split}$
(14)

Equation (11) can then be modified to calculate a unified $p_{\mathrm{astro}}$
marginalized over signal and noise counts,

$p_{\rm
astro}(\vec{x})=\int\mathrm{d}\Lambda_{\mathrm{s}}{}{}\,\mathrm{d}\Lambda_{\mathrm{n}}{}{}\frac{\Lambda_{\mathrm{s}}{}{}p(\vec{x}|\mathcal{S})\,p(\Lambda_{\mathrm{s}}{}{},\Lambda_{\mathrm{n}}{}{}|\\{\vec{x}\\},{N})}{\Lambda_{\mathrm{s}}{}{}p(\vec{x}|\mathcal{S})+\Lambda_{\mathrm{n}}{}{}p(\vec{x}|\varnothing)}.$
(15)

This equation therefore incorporates correlations between pipelines through
the joint likelihoods $p(\vec{x}|\mathcal{S})$ and $p(\vec{x}|\varnothing)$.

The role of correlations in calculating $p_{\mathrm{astro}}$ may be understood
by considering a few idealized cases. If we had two pipelines that looked for
the same type of signal, but had different sensitivities to noise triggers, we
might expect to be more certain in a candidate (assign a higher
$p_{\mathrm{astro}}$) if there are corresponding triggers from both pipelines.
However, if a candidate was identified by one pipeline, and not by another
which has greater sensitivity to that type of signal, we might expect to
suspect it as a false alarm (assign a lower $p_{\mathrm{astro}}$). In the
following subsections, we will demonstrate the properties of our unified
$p_{\mathrm{astro}}$ in some illustrative cases.

##  IV Toy Model

In this section we illustrate and test the unified $p_{\mathrm{astro}}$ method
using a simple toy model. In Sec. V, we will then apply the method on real GW
data, albeit in a simplified analysis.

The toy model data is generated in the form of segments, each of which
consists of four data points with values drawn from a standard normal
distribution. A segment might also contain a signal with a probability
proportional to $\Lambda_{\mathrm{s}}{}{}$. Signals are simulated by adding a
value $\lambda_{\mathrm{s}}$ to one of the data points in the segment. We
assume that $\lambda_{\mathrm{s}}$ is known. The goal of an FGMC-type analysis
is to estimate the fraction $\Lambda_{\mathrm{s}}{}{}$. Figure 1 demonstrates
a schematic of the data generated in the toy model.

We construct four simple pipelines to analyze this data on a segment-by-
segment basis. The pipelines are all based on a maximum-likelihood estimate
assuming Gaussian noise statistics, and the pipelines all assume that the
noise is drawn from a standard normal with $\sigma=1$. For each segment, the
pipelines calculate the noise likelihood

$\mathcal{L}_{\mathrm{n}}\propto\exp\left[-\frac{1}{2}\frac{\left(\sum_{i=1}^{k}x_{i}\right)^{2}}{k\sigma^{2}}\right],$
(16)

and the signal likelihood, assuming the signal value $\lambda_{\mathrm{s}}$ is
known

$\mathcal{L}_{\mathrm{s}}\propto\exp\left[-\frac{1}{2}\frac{\left(\sum_{i=1}^{k}x_{i}-\lambda_{\mathrm{s}}\right)^{2}}{k\sigma^{2}}\right],$
(17)

where $x_{i}$ are the analyzed data points in the segment. The pipelines might
use different number of data points, corresponding to the number $k\leq 4$.
Each pipeline calculates a p value for every segment (under the noise
hypothesis) which is the main input for the unified $p_{\mathrm{astro}}$
analysis.

The four pipelines are

  1. 1.

Pipeline 1: This pipeline makes no assumption on which of the four data points
in a segment has the signal, effectively marginalizing over the position of
the signal. Therefore, $k=4$ in Eq. (16) and Eq. (17).

  2. 2.

Pipeline 2: This pipeline still assumes Gaussian statistics but only considers
the data point with the loudest value of the four and calculates its
likelihood. The is equivalent to setting $k=1$ in Eq. (16) and Eq. (17), but
replacing $x_{i}$ with $\max\\{x_{i}\\}$. This pipeline will be more effective
at detecting a loud signal but will generally overestimate the number of
signals.

  3. 3.

Pipeline 3: This only uses the first two data points of a segment. Therefore,
$k=2$ in Eq. (16) and Eq. (17). Since in our toy model, we know that each data
point is equally probable to contain a signal, we expect that this pipeline
will witness only about half of the total signals, and consequently its count
parameter will be around half the true value. Thereby, this is analogous to a
search pipeline that will be sensitive to only a subset of GW signals (e.g.,
those from high-mass systems).

  4. 4.

Pipeline 4: Similar to Pipeline 3 except it only uses the last two data
points. Therefore Pipeline 4 is statistically independent from Pipeline 3
while still seeing only half of the total signals. Combining this pipeline
with Pipeline 3 is the simplest example of a correlation between pipelines.

For each pipeline, we also estimate the single-pipeline signal and noise
counts using the signal and noise likelihoods.

Figure 1: A schematic of the toy model data. Each row is a segment, consisting
of four data points indicated by the rounded rectangles. The noise in the
segments is drawn from a standard normal distribution, with its value
indicated by the color scale of the rounded rectangle. Segments 2 and 4 from
the top in this example contain a signal, added randomly to one of the four
data points indicated by the vermilion star.

Using the p value from the pipelines as their detection statistics, we will
calculate the joint likelihoods using a KDE as described in Sec. III. To do
this we create a number of segments containing a signal to train a signal KDE,
and similarly train a noise KDE using segments that do not contain a signal.
Then for a simulation where $\Lambda_{\mathrm{s}}$ is unknown, we can score
the outputs of the pipelines against the KDEs to perform a unified FGMC
analysis using Eq. (14), and measure the noise and signal counts,
$\Lambda_{\mathrm{n}}{}{}$ and $\Lambda_{\mathrm{s}}{}{}$.

(a) Analysis with Pipeline 1 and Pipeline 2. The horizontal and vertical axes
are the measured signal and noise count parameters. Pipeline 2 is clearly
biased as it only uses the loudest data point of a segment for analysis, but
the unified FGMC analysis performing a joint fit can correct for it.

(b) Analysis with Pipeline 3 and Pipeline 4. While both pipelines only see
half the simulated signals, the joint analysis learns to correct for this from
the correlations in the joint signal distributions.

Figure 2:  Toy-model exploration of the unified $p_{\mathrm{astro}}$
formalism. A signal value, $\lambda_{\mathrm{s}}=3$ was used in both cases. In
both figures, the solid vertical and horizontal lines indicate the real number
of segments with (and without) a signal. The shaded regions in the marginal
plots are $90\%$ uncertainty levels while the two-dimensional contours
correspond to 50% and 90% levels.

The algorithmic choices made by the pipelines mean that their results differ
from each other, and can introduce a bias in their estimate of signal counts
(and thereby $p_{\mathrm{astro}}$). For instance, consider Pipeline 3, which
can only see about half of the total signals. When combining the output of the
pipelines, the KDEs do not know a priori about these biases and will have to
learn it from the triggers. In a joint analysis between Pipeline 3 and
Pipeline 4, the signal KDE can learn that only about half the analyzed
segments will have a low p value in Pipeline 3 (i.e., when a signal is added
to the first two data points) and that these are the segments in which
Pipeline 4 will likely give a high p value (and vice versa).

Figure 2 shows an FGMC analysis for two separate toy model analyses. In both
Figs. 2(a) and 2(b), we separately simulated $1000$ segments of data with
about a quarter ($\Lambda_{s}=250$) of the segments having a signal with
$\lambda_{\mathrm{s}}=3$. Figure 2(a) shows the signal and noise counts
estimated by Pipeline 1 and Pipeline 2, along with a unified fit. As expected,
Pipeline 2 shows a bias in its recovery which nevertheless the joint analysis
can correct for. Figure 2(b) shows the results of a toy-model analysis with
Pipeline 3 and Pipeline 4. Both pipelines only see about half of the simulated
signals, by design. However, the joint analysis learns this bias through the
signal KDE and can account for it.

This section shows, using a simple example, how outputs from multiple
pipelines can be combined using the unified formalism that we have developed.
It also shows how certain kinds of biases can be corrected when multiple
pipelines are joined together, by drawing information from the correlations
(or lack thereof) between the outputs of the pipelines. These examples
demonstrate the robustness of the unified $p_{\mathrm{astro}}$ method.

(a) The signal distribution $p(\vec{x}|\mathcal{S})$ learned using a KDE to
the simulated signals that are commonly found by both GstLAL and PyCBC. The
stars are the O3a on-source triggers from GWTC-2.1 found by both pipelines
Abbott  _et al._ (2021d). The upper cutoff of the PyCBC distribution that is
visible in the plot comes from the fact that the pipeline places a limit on
the FAR based on the number of time slides performed. This translates to a
cutoff at $\beta\approx 15.76$ and results in a bump in the PyCBC distribution
at these values of $\beta$.

(b) The noise distribution $p(\vec{x}|\varnothing)$ fit to a $2$-dimensional
Gaussian to the common noise triggers. The stars are the O3a on-source
triggers from GWTC-2.1 found by both pipelines Abbott  _et al._ (2021d).

Figure 3: Joint likelihood distributions for the signal and noise hypothesis.
The spans of the signal and noise distributions span are dissimilar. The
signal distribution extends to much larger $\beta$ values than the noise
distribution. The contours correspond to $50\%$ and $90\%$ levels in both
plots. Figure 4: The distribution of triggers that were found by one pipeline
only. The top plot shows the noise and signal distribution of GstLAL, while
the bottom plot shows the PyCBC distribution of triggers. The black stars are
again the on-source O3a triggers from GWTC-2.1 that have only been found by
the corresponding pipeline Abbott  _et al._ (2021d).

##  V Application to GWTC-2.1

We now test the unified $p_{\mathrm{astro}}$ method with real GW triggers
(henceforth _on-source_ triggers) by applying this to O3a results from
GWTC-2.1 Abbott  _et al._ (2021e). GWTC-2.1 applies a preliminary cut of
$\mathrm{FAR}\leq 2~{}\mathrm{day}^{-1}$, which we also adopt. While this
analysis uses real data, it is only intended to be illustrative, and we adopt
some simplifications for computational simplicity.

We use results from the GstLAL Messick  _et al._ (2017); Sachdev  _et al._
(2019); Cannon  _et al._ (2021) and PyCBC Usman  _et al._ (2016); Nitz  _et
al._ (2017); Davies  _et al._ (2020) pipelines, and restrict ourselves to
triggers that are at least Hanford-Livingston coincident (i.e., present in
data from LIGO Hanford and Livingston at a minimum). GWTC-2.1 Abbott  _et al._
(2021e) uses two versions of the PyCBC pipelines; referred to as PyCBC-broad
and PyCBC-BBH; in this analysis we only use the former version, which we will
simply refer to as PyCBC.111Our approach could be used to combine the PyCBC-
broad and PyCBC-BBH results too; we pick the GstLAL and PyCBC-broad results to
illustrate our method to show how it can be used across different pipelines.
While our formalism is general, we focus solely on CBC signals; a choice that
is primarily driven by prudence, as we have only seen the population of CBC
signals so far Abbott  _et al._ (2021a). Furthermore, for the sake of
simplicity we will only use BBH signals for the training; this will also serve
as a test to see if the method can distinguish sufficiently between signal
types and to test overfitting. These simplifications make it easier to
illustrate the behavior of the unified $p_{\mathrm{astro}}$, and highlight the
key considerations for an analysis to be performed for future GW catalogs.

We use pipeline settings that are as similar as possible to the runs in
GWTC-2.1 Abbott  _et al._ (2021e). A crucial factor to consider is that the
search pipelines have to be run in a way that we can establish a one-to-one
correspondence between their triggers so that we can learn the correlations
between them. This is necessary even in the case where one pipeline registers
a trigger and the other does not because the absence of the trigger is itself
useful information. While one-to-one correspondence is straightforward for
simulated signal triggers (described in Sec. V.1), it can be somewhat
ambiguous to define in the case of noise triggers (implementation details are
explained in Sec. V.2).

In order to calculate a unified $p_{\mathrm{astro}}$, Eq. (14), and Eq. (15)
require us to construct the joint distribution of a statistic from each
pipeline. We choose to use the FAR as the base statistic here as it is easily
interpretable, commonly used by all pipelines and informative of the relative
significance of triggers. However, the dynamic range of FARs can be large, and
we therefore define a statistic $\beta$ that is related to the logarithm of
the inverse FAR,

$\beta(\mathrm{FAR})=\log\left(1+\frac{\kappa}{\mathrm{FAR}}\right),$ (18)

where $\kappa$ is a scaling constant, which we set to a value of
$\kappa=100~{}\mathrm{yr}^{-1}$. This number is set to give a good dynamic
range to $\beta$ but our results are insensitive to its actual value. Higher
values of $\beta$ correspond to more significant candidates.

We construct three cases each for the noise and signal hypotheses,
corresponding to triggers that are registered (i) only by GstLAL, (ii) only by
PyCBC, and (iii) by both pipelines. We find that separate modeling the three
cases is necessary in order to achieve a faithful fit of signal triggers,
which can cover a wide range on the $\beta$ space and show significant
structure. In addition, this allows us to consider the case where one pipeline
would not see a signal but the other does. For example, in the GWTC-2.1
analysis Abbott  _et al._ (2021d), PyCBC did not assign FARs to events which
triggered in a single detector (although this is now possible Davies and Harry
(2022)), but GstLAL did. In the future, a more sophisticated fitting method
like a convolutional neural network or a random forest classifier might make
multiple separate fits unnecessary.

The distributions are usually normalized such that the total probability over
its parameter space is one. Therefore, when considering multiple separate
cases we need to renormalize them to account for the relative probability of
the class they represent. We do this by multiplying the output of the
distribution with the fraction of signal or noise triggers that fall in that
particular class. We find that getting a joint trigger is $\approx 104$ times
more likely under the signal hypothesis than the noise hypothesis. Meanwhile,
a GstLAL-only (PyCBC-only) trigger is $\approx 2.77$ ($\approx 3.87$) times
more likely under the noise hypothesis than the signal hypothesis. The numbers
quoted correspond to triggers that pass a $2~{}\mathrm{day}^{-1}$ FAR
threshold.

Figure 5: The posterior for the signal and noise counts in the joint analysis.
The shaded region in the one-dimensional posterior corresponds to $90\%$
uncertainty levels, while the contours in the two-dimensional posteriors are
the $50\%$ and $90\%$ levels. We recover a median value
$\Lambda_{s}=53^{+10}_{-13}$.

###  V.1 Signal distribution

The signal distributed is generated by fitting simulated software signals or
_injections_ that are added to the detector noise, to a KDE. We use the same
simulated signals as from the GWTC-3 analysis Collaboration  _et al._ (2021),
whose distribution is described in detail in Appendix C.7 (the injection row
of Table X) of the GWTC-3 paper Abbott  _et al._ (2021a). In particular, the
source masses are distributed as

$\begin{split}p(m_{1})&\propto m_{1}^{-2.35},\\\ \quad
p(m_{2}\,|\,m_{1})&\propto m_{2}.\end{split}$ (19)

The redshift distribution is flat in comoving volume, with a maximum redshift
of $1.9$, assuming a $\Lambda$CDM cosmology Ade  _et al._ (2016). We restrict
ourselves to injections with source-frame component masses
$3M_{\odot}$–$100M_{\odot}$ to focus solely on BBHs. The simulated population
is not a close match to the inferred astrophysical population Abbott  _et al._
(2021b); Callister and Farr (2023). However, it shares broad characteristics
and is simple to use. Therefore it suffices for our illustrative calculation.
Using a population model that more closely resembles the underlying population
should enable more accurate calculation of the true alarm rate, and hence
$p_{\mathrm{astro}}$.

In any KDE-based analysis, the choice of the KDE bandwidth is important,
especially for multidimensional distributions with intricate shapes. In our
illustrative analysis, the bandwidth was picked manually by checking that the
KDE approximates well the distribution of injections. We verified that the
one-dimensional cumulative distributions of the KDE and the injection set
match, and also cross-checked the KDE against a KDE using the Silverman rule
of thumb Silverman (1986). Ultimately, a realistic application would need a
more sophisticated fitting scheme, for example, KDEs fit using an iterative
approach Sadiq  _et al._ (2023) or a machine-learning approach such as using
normalizing flows.

The top plot of Fig. 3(a) shows the joint distribution of signals that are
found by both PyCBC and GstLAL, and Fig, 4 shows the distribution of signal
triggers that are found only by one pipeline.

###  V.2 Noise distribution

The noise distribution is fit using data with a nonphysical time shift higher
than the light travel time between detectors to remove correlations between
any real signals. Generating this data is computationally expensive,
therefore, in this proof-of-concept analysis, we use O3a data with a single
time shift where LIGO Livingston and Virgo data streams were shifted by
$0.62831$ and $0.31415~{}\mathrm{s}$, respectively, in relation to LIGO
Hanford.

To define corresponding noise triggers between the pipelines, we match the
times of the time-shifted triggers within a window of $1~{}\mathrm{s}$,
comparable to the typical window used for grouping triggers in O3 Abbott  _et
al._ (2021a). This procedure gives us $256$ GstLAL triggers and $348$ PyCBC
triggers that pass the FAR threshold of $2~{}\mathrm{day}^{-1}$; of these,
only $12$ triggers are common between pipelines. For a unified
$p_{\mathrm{astro}}$ analysis intended to be used in production of a GW
catalog, a larger set of noise triggers would be required.

The small number of noise triggers common between the pipelines makes it
difficult to accurately reconstruct the distribution. However, as our primary
goal is to illustrate unified $p_{\mathrm{astro}}$ method, rather than to
produce a full set of results, these triggers are sufficient for completing a
demonstrative calculation. To make use of the small number of triggers that
are common between the pipelines, we adopt a bootstrapping procedure to
generate more triggers. To do this, we first fit triggers that pass a
$10~{}\mathrm{day}^{-1}$ cut to a KDE, and draw $10^{4}$ triggers that pass a
$2~{}\mathrm{day}^{-1}$ cut. This corresponds to doing $\approx 17$ time
slides in total. To avoid overfitting the noise triggers, we choose to employ
a truncated Gaussian fit to them, instead of a KDE. The assumption of a
Gaussian is purely for simplicity, and a more careful choice of estimating the
noise distribution would be needed for a proper analysis. The single-pipeline
noise triggers are fit using a truncated Gaussian with mean
$\mu=\beta(2~{}\mathrm{day}^{-1})$ and a standard deviation calculated from
the noise triggers. The joint noise distribution is fit with a $2$-dimensional
truncated Gaussian with $\mu=\beta(2~{}\mathrm{day}^{-1})$ and the covariance
matrix calculated from the noise triggers. Figure 3(b) shows the distribution
of joint triggers, while Fig. 4 shows the single-pipeline triggers in this
bootstrapped data set.

The sparseness of noise triggers means that the noise distribution is not
accurately reconstructed. This is liable to give biased values for
$p_{\mathrm{astro}}$: where the noise likelihood is underestimated,
$p_{\mathrm{astro}}$ will be underestimated, and where the noise likelihood is
overestimated, $p_{\mathrm{astro}}$ will be underestimated. However, the goal
of this application is to show that the unified $p_{\mathrm{astro}}$ formalism
yields sensible results to illustrate what would be needed for future
analyses. A realistic application of this process to GW data will likely
require multiple time shifts or some other process to generate a larger number
of high-fidelity noise triggers, and then a careful reconstruction of the
shape of the noise distribution.

###  V.3 Computing a unified $p_{\mathrm{astro}}$

|  | GstLAL | PyCBC |
---|---|---|---|---
GPS time | GW name | FAR $(\text{yr}^{-1})$ | $\beta$ | FAR $(\text{yr}^{-1})$ | $\beta$ | Unified $p_{\rm astro}$
$1238782700.0$ | GW$190408\\_181802$ | $2.1\times 10^{-15}$ | $38.4$ | $2.5\times 10^{-04}$ | $12.9$ | $>0.99$
$1239082262.0$ | GW$190412\\_053044$ | $1.9\times 10^{-27}$ | $66.1$ | $1.1\times 10^{-04}$ | $13.7$ | $>0.99$
$1240327333.0$ | - | $9.1\times 10^{-01}$ | $4.7$ | $4.2\times 10^{+01}$ | $1.2$ | 0.98
$1240944862.0$ | GW$190503\\_185404$ | $2.3\times 10^{-06}$ | $17.6$ | $3.8\times 10^{-02}$ | $7.9$ | $>0.99$
$1241108686.0$ | - | $1.7\times 10^{+01}$ | $1.9$ | - | - | 0.85
$1241719652.0$ | GW$190512\\_180714$ | $7.7\times 10^{-12}$ | $30.2$ | $1.1\times 10^{-04}$ | $13.7$ | $>0.99$
$1241816086.0$ | GW$190513\\_205428$ | $1.3\times 10^{-05}$ | $15.8$ | - | - | $>0.99$
$1242107479.0$ | GW$190517\\_055101$ | $4.5\times 10^{-03}$ | $10.0$ | $9.4\times 10^{-03}$ | $9.3$ | $>0.99$
$1242315362.0$ | GW$190519\\_153544$ | $2.2\times 10^{-06}$ | $17.6$ | $1.0\times 10^{-04}$ | $13.8$ | $>0.99$
$1242442967.0$ | GW$190521\\_030229$ | $2.0\times 10^{-01}$ | $6.2$ | $4.4\times 10^{-01}$ | $5.4$ | $>0.99$
$1242459857.0$ | GW$190521\\_074359$ | $5.0\times 10^{-33}$ | $79.0$ | $1.8\times 10^{-05}$ | $15.5$ | $>0.99$
$1242984073.0$ | GW$190527\\_092055$ | $2.3\times 10^{-01}$ | $6.1$ | - | - | $>0.99$
$1243533585.0$ | GW$190602\\_175927$ | $1.1\times 10^{-07}$ | $20.6$ | $2.9\times 10^{-01}$ | $5.9$ | $>0.99$
$1243926576.0$ | - | $1.2\times 10^{+01}$ | $2.2$ | $6.4\times 10^{+02}$ | $0.1$ | 0.72
$1243985856.0$ | - | $3.2\times 10^{+02}$ | $0.3$ | $2.6\times 10^{+02}$ | $0.3$ | 0.59
$1245221073.0$ | - | $1.2\times 10^{+02}$ | $0.6$ | $1.8\times 10^{+02}$ | $0.4$ | 0.53
$1245874666.0$ | - | $6.2\times 10^{+02}$ | $0.1$ | $2.2\times 10^{+02}$ | $0.4$ | 0.80
$1246048404.0$ | GW$190701\\_203306$ | $5.7\times 10^{-03}$ | $9.8$ | $6.4\times 10^{-02}$ | $7.4$ | $>0.99$
$1246385767.0$ | - | $5.3\times 10^{+00}$ | $3.0$ | - | - | $>0.99$
$1246487219.0$ | GW$190706\\_222641$ | $5.0\times 10^{-05}$ | $14.5$ | $3.7\times 10^{-04}$ | $12.5$ | $>0.99$
$1246779793.0$ | - | $1.0\times 10^{+01}$ | $2.4$ | - | - | 0.99
$1246849694.0$ | - | $2.7\times 10^{+00}$ | $3.6$ | - | - | $>0.99$
$1247616534.0$ | GW$190720\\_000836$ | $4.4\times 10^{-08}$ | $21.5$ | $1.4\times 10^{-04}$ | $13.5$ | $>0.99$
$1248242631.0$ | GW$190727\\_060333$ | $2.7\times 10^{-10}$ | $26.6$ | $5.6\times 10^{-03}$ | $9.8$ | $>0.99$
$1248331528.0$ | GW$190728\\_064510$ | $5.4\times 10^{-16}$ | $39.8$ | $8.2\times 10^{-05}$ | $14.0$ | $>0.99$
$1248617394.0$ | GW$190731\\_140936$ | $3.3\times 10^{-01}$ | $5.7$ | - | - | $>0.99$
$1249479778.0$ | - | $6.6\times 10^{+00}$ | $2.8$ | - | - | $>0.99$
$1250620378.0$ | - | $5.2\times 10^{+00}$ | $3.0$ | - | - | $>0.99$
$1251009263.0$ | GW$190828\\_063405$ | $5.0\times 10^{-27}$ | $65.2$ | $8.5\times 10^{-05}$ | $14.0$ | $>0.99$
$1251010527.0$ | GW$190828\\_065509$ | $3.5\times 10^{-05}$ | $14.9$ | $2.8\times 10^{-04}$ | $12.8$ | $>0.99$
$1251588283.0$ | - | $1.6\times 10^{+01}$ | $2.0$ | - | - | 0.92
$1251926900.0$ | - | $1.0\times 10^{+01}$ | $2.4$ | - | - | $>0.99$
$1252627040.0$ | GW$190915\\_235702$ | $7.8\times 10^{-06}$ | $16.4$ | $6.8\times 10^{-04}$ | $11.9$ | $>0.99$
$1252699636.0$ | GW$190916\\_200658$ | $1.2\times 10^{+01}$ | $2.2$ | - | - | 0.99
$1252756008.0$ | GW$190917\\_114630$ | $6.6\times 10^{-01}$ | $5.0$ | - | - | $>0.99$
$1252939489.0$ | - | $9.0\times 10^{+00}$ | $2.5$ | - | - | $>0.99$
$1252987339.0$ | - | $5.1\times 10^{+01}$ | $1.1$ | $8.0\times 10^{+01}$ | $0.8$ | 0.65
$1253326744.0$ | GW$190924\\_021846$ | $5.0\times 10^{-10}$ | $26.0$ | $8.2\times 10^{-05}$ | $14.0$ | $>0.99$
$1253509434.0$ | GW$190926\\_050336$ | $1.1\times 10^{+00}$ | $4.5$ | - | - | $>0.99$
$1253755327.0$ | GW$190929\\_012149$ | $1.5\times 10^{-01}$ | $6.5$ | $1.2\times 10^{+02}$ | $0.6$ | $>0.99$
$1253885759.0$ | GW$190930\\_133541$ | $4.3\times 10^{-01}$ | $5.5$ | $1.8\times 10^{-02}$ | $8.6$ | $>0.99$
Table 1: Triggers with a unified $p_{\mathrm{astro}}{}\geq 0.5$ from our
illustrative analysis. The triggers that have $p_{\mathrm{astro}}{}{}\geq 0.5$
in at least one pipeline in GWTC-2.1 Abbott  _et al._ (2021d) are shown in the
second column. Also listed are the FARs of the triggers from the GstLAL and
PyCBC pipelines. The GW name of the triggers also recovered with
$p_{\mathrm{astro}}$$\geq 0.5$ by at least one pipeline in GWTC-2.1 is given
in the second column. These results illustrate the properties of the unified
$p_{\mathrm{astro}}$ method, but a larger number of noise triggers, and more
accurate population models, would be needed to obtain reliable quantitative
results.

After restricting ourselves to ones that are at least Hanford-Livingston
coincident, and have been found by at least one of GstLAL or PyCBC, we are
left with $553$ on-source triggers from the GWTC-2.1 O3a results Abbott  _et
al._ (2021a). Using the signal and noise models described above, we can now
proceed to calculate a unified $p_{\mathrm{astro}}$ for these triggers.

First, through Eq. (10), we estimate $\Lambda_{\mathrm{s}}{}{}$ and
$\Lambda_{\mathrm{n}}{}{}$ for the joint analysis. We do this using the
Markov-chain Monte Carlo sampler emcee Foreman-Mackey  _et al._ (2013). We use
uniform priors on the count parameters, $\Lambda_{\mathrm{s}}{}{}\in[0,1000]$
and $\Lambda_{\mathrm{n}}{}{}\in[0,1000]$. The triggers are then scored
against the appropriate distribution to calculate $p({x_{i}}|\mathcal{S})$ and
$p({x_{i}}|\varnothing)$.

Figure 5 shows the posterior distribution of $\Lambda_{\mathrm{s}}{}{}$ and
$\Lambda_{\mathrm{n}}{}{}$, with a median $\Lambda_{\mathrm{s}}{}{}$ of $53$.
Table 1 lists the $41$ triggers that have unified $p_{\mathrm{astro}}{}\geq
0.5$. Most triggers are reported by both GstLAL and PyCBC, but we do have a
few triggers that are detected by GstLAL alone, which corresponds to the high
$\beta$ tail of the on-source triggers in the top panel of Fig. 4. This is
qualitatively consistent with the $44$ triggers found with a
$p_{\mathrm{astro}}{}\geq 0.5$ by at least one pipeline in GWTC-2.1 analysis
Abbott  _et al._ (2021a).

In Table 1, we recover $26$ triggers that are also reported with
$p_{\mathrm{astro}}{}{}>0.5$ in at least one search pipeline in the GWTC-2.1
analysis Abbott  _et al._ (2021a) (the corresponding GW identification of
these triggers has been given for easy identification) while the remaining
$15$ triggers were below this threshold. Inspecting the $\beta$ values of
these triggers show that they are usually low, and comparing with the noise
distributions, it is plausible that they could be consistent with noise if
some of the modeling assumptions are relaxed. The simplified modeling of the
noise distribution is expected to lead to the promotion of some noise triggers
while suppressing some real signals.

|  | GstLAL | PyCBC |
---|---|---|---|---
GPS time | GW name | FAR $(\text{yr}^{-1})$ | $\beta$ | FAR $(\text{yr}^{-1})$ | $\beta$ | Unified $p_{\rm astro}$
$1239917954.0$ | GW$190421\\_213856$ | $2.8\times 10^{-03}$ | $10.5$ | $5.9\times 10^{+00}$ | $2.9$ | $<0.01$
$1246527224.0$ | GW$190707\\_093326$ | $2.7\times 10^{-15}$ | $38.2$ | $9.7\times 10^{-06}$ | $16.1$ | $<0.01$
$1248834439.0$ | GW$190803\\_022701$ | $7.3\times 10^{-02}$ | $7.2$ | $8.1\times 10^{+01}$ | $0.8$ | $<0.01$
Table 2: Triggers with a unified $p_{\mathrm{astro}}{}<0.5$ in our
illustrative analysis, but with a $p_{\mathrm{astro}}{}\geq 0.5$ in either
GstLAL or PyCBC in GWTC-2.1 Abbott  _et al._ (2021d)

In the case of joint triggers there is a heavy weight in favor of signal
triggers as discussed in Sec. V. The exact quantitative results might be
susceptible to small-number statistics, and it is plausible that some of these
would be down weighted if we had a more complete noise distribution from more
time slides. However, the results show the expected behavior that having
multiple pipelines find a candidate when their noise distributions are largely
uncorrelated, increases our certainty that a candidate is real.

In Table 2, we show the list of triggers with a unified
$p_{\mathrm{astro}}{}{}<0.5$, but with a $p_{\mathrm{astro}}{}{}\geq 0.5$ in
either GstLAL or PyCBC in GWTC-2.1 Abbott  _et al._ (2021d). While the
correlation between pipelines increases $p_{\mathrm{astro}}$ for some
candidates, the unified analysis also down-weights certain triggers, in
particular those with highly asymmetric $\beta$ values. This is because the
joint simulated-signal trigger distribution has a strong correlation between
the $\beta$ (and thereby the FAR) values of two pipelines; this is expected
since both are matched filter pipelines using CBC templates. For example,
GW190803_022701 has GstLAL and PyCBC FARs of $7.3\times
10^{-2}~{}\mathrm{yr}^{-1}$ and $81~{}\mathrm{yr}^{-1}$, respectively.
Therefore, while the GstLAL significance of the trigger is high, the
asymmetric FARs in the two pipelines gives it a lower weight in the unified
analysis. In our case, the low $p_{\mathrm{astro}}$ can potentially be
explained by the choice of a Gaussian for the noise distribution with heavy
tails at large $\beta$. The uncertainties involved in the noise and the signal
distributions in this illustrative analysis, and the fact that we do not
include other search pipelines, mean that we should not rule out the triggers
in Table 2. Nevertheless, the results demonstrate that there is information
that is uniquely captured by a joint analysis of multiple pipelines. In
practice, we would expect that any high-significance candidate that is a clear
outlier in the signal distribution would warrant further investigation to
understand why the pipelines differ in their response.

Finally, many of the triggers in Table 1 have a $p_{\mathrm{astro}}$ close to
$1$. This is likely an overestimate, due to the sparseness of the noise
distribution which can lead to an artificially low noise likelihood in Eq.
(15). We equally expect that some of the low-probability candidates, not shown
in the table, have underestimated $p_{\mathrm{astro}}$ for the same reason.
The specific values we get are also dependent on the strong modeling
assumptions made for the noise distribution. A more realistic analysis would
probably require adequate non-parametric modeling of the noise distribution
such that any structure in the parameter space can be identified. While it is
worth bearing these limitations in mind, both Table 1 and Fig. 5 qualitatively
demonstrate consistency of the results with GWTC-2.1, indicating that even
with simpler modeling choices this method can yield sensible overall results.

##  VI Discussion and conclusion

We have developed a statistical formalism for combining information from
multiple search pipelines to calculate a unified $p_{\mathrm{astro}}$. We
first demonstrated this formalism using a simple toy model, showing that it
can consistently combine information and even account for biases in search
pipelines. We then applied the framework to O3a data, using triggers from
GstLAL and PyCBC, demonstrating how correlations between pipelines may update
our understanding of candidates, but highlighting the importance of using
accurate models for the signal and noise populations.

Currently, multiple search pipelines are used to identify interesting
candidates, each with its own strengths. A unified $p_{\mathrm{astro}}$ can
potentially reduce confusion in the interpretation of triggers found by
different pipelines and can help in using marginal triggers for subsequent GW
analyses. We have shown that certain types of correlations between pipelines
can inform the significance that the unified analysis assigns to a trigger.
Similarly, combining the search pipelines’ results mitigates the need to
calculate an effective trials factor to correct the individual-pipeline FARs
to account for repeated analysis of the same data. Calculating this trials
factor is generally nontrivial. Our formalism can naturally and consistently
account for these considerations since it calculates how triggers in one
pipeline are correlated with those in another pipeline.

While we have demonstrated that this method can estimate signal and noise
counts, and a unified $p_{\mathrm{astro}}$ in a sensible and consistent
manner, the paucity of noise triggers is a clear computational bottleneck. The
scarcity of these triggers can be understood by considering the FAR thresholds
used. The $2~{}\mathrm{day}^{-1}$ threshold used means that we will be limited
to a few hundred noise triggers in each pipeline per a six month run.
Therefore, depending on the structure in the joint noise correlations, we
would probably need $\mathcal{O}(10$–$100)$ time slides to obtain an accurate
representation of the noise distribution. The number of noise triggers that
are common between pipelines will be only a fraction of the total number of
noise triggers, and if the noise backgrounds are distinct, then there will
only be a small fraction in common. This may indicate a need for multiple
coordinated time shifts between pipelines to build up the noise distribution.
One can also consider other methods of building noise distributions such as
modeling the joint noise distribution and drawing from it, similar in
philosophy to the GstLAL pipeline Cannon  _et al._ (2013).

However, while the low number of common noise triggers does pose a difficulty
for reconstructing the shape of the noise distribution, it also highlights the
importance of considering the number of candidates in common between
pipelines: if common noise triggers are rare, but common signal triggers are
frequent, then a candidate being found by multiple pipelines should increase
its $p_{\mathrm{astro}}$. We already see this in the analysis done here where
a joint trigger is about $100$ times more likely under the signal hypothesis.

If suitable data products (injection sets and noise triggers) are available,
this formalism can be readily applied to include other search pipelines,
notably those used by the LVK such as MBTA Adams  _et al._ (2016); Aubin  _et
al._ (2021) and cWB Klimenko and Mitselmakher (2004); Klimenko  _et al._
(2011, 2016), as well as pipelines developed for external analysis of public
data Venumadhav  _et al._ (2019). This would enable the construction of a
single GW catalog accounting for all analyses. A strength of this formalism
when constructing joint catalogs is that it would differentially weigh search
pipelines, depending on their precision and accuracy by taking into account
the shape of the joint simulated-signal distribution. Hence, as long as the
simulated-signal distribution sufficiently reflects the true distribution of
GW sources, it should be possible to correct for pipelines missing signals
over some region of parameter space, or producing spurious triggers in
another.

The addition of cWB (and other minimally modeled pipelines Lynch  _et al._
(2017); Kanner  _et al._ (2016)) to catalog production could be extremely
useful, as it can be sensitive in regions of the CBC parameter space where
modeled searches do not perform well, such as eccentric binaries Abbott  _et
al._ (2019b), in addition to non-CBC transient sources. The sensitivity to
non-CBC sources is a complication for current analyses, as the
$p_{\mathrm{astro}}$ calculations are done assuming only CBC sources
(discussed in Appendix F of the GWTC-3 paper Abbott  _et al._ (2021c)). Non-
CBC source have a lower true alarm rate than CBC sources, and hence a lower
$p_{\mathrm{astro}}$ at a given FAR; consequently, misidentifying a trigger as
CBC will lead to an overestimate of its $p_{\mathrm{astro}}$. To mitigate
this, LVK analyses have imposed an additional criterion for cWB candidates,
that they must have a counterpart trigger from a template-based CBC pipeline
Abbott  _et al._ (2019a, 2021a). This is effectively an approximation to the
unified $p_{\mathrm{astro}}$ framework, acknowledging that for real CBC
signals there would probably be correlation between pipeline results. This
could be put on a more rigorous basis by explicitly using out unified
$p_{\mathrm{astro}}$ framework, considering the response of the pipelines to
simulated CBC and non-CBC signals.

Addition of more pipelines can make the problem of noise triggers more acute,
as the number of triggers that are common between three or more pipelines
might be small and extrapolation difficult. However, it is probable that noise
triggers in a multiple-pipeline space would be so rare that we can
conservatively set the noise likelihood to a small limiting value for such a
case.

In this paper, we have only considered the simplest case of calculating a
unified $p_{\mathrm{astro}}$. In a future work, we will extend it to implement
binning in the mass space to estimate the astrophysical probability that a
system is a BBH, a BNS or a neutron star–black hole binary (NSBH) Kapadia  _et
al._ (2020). Such an extension is in principle straightforward, where in
addition to a statistic like the FAR we also fit distribution of recovered
template chirp masses for simulated BBH, BNS and NSBH signal. On-source
triggers can then be scored against such two-dimensional distributions to give
the corresponding likelihood. Searches are often optimized in different ways
giving them differential sensitivity in specific parts of the mass space
(e.g., PyCBC has an analysis tuned to BBHs Nitz  _et al._ (2020); Abbott  _et
al._ (2021c)). Calculating unified probabilities in conjugation with mass
binning will allow us to fold in this differential sensitivity in a
consistent, unbiased way in a way akin to the toy model example with Pipeline
3 and Pipeline 4 in Sec. IV.

A key uncertainty in the calculation of $p_{\mathrm{astro}}$ is the form of
the underlying source population. Errors in the assumed population translate
to a misestimation of the true alarm rate, and hence $p_{\mathrm{astro}}$. A
way to mitigate this would be to infer the population simultaneously while
calculating $p_{\mathrm{astro}}$ Gaebel  _et al._ (2019); Galaudage  _et al._
(2020); Roulet  _et al._ (2020). This additionally enables lower-significance
candidates to be used in population inference. Using a unified
$p_{\mathrm{astro}}$ makes it easier to assess the contamination fraction in a
catalog based upon several pipelines, and hence would make it more convenient
to perform such joint inference in the future.

While we have only considered the application of our method to final search
analyses performed offline, an extension to low-latency detections is also
theoretically possible. By combining multiple pipelines we depend less on one
pipeline and hence should be less susceptible to incorrect alerts and
retractions. In combination with mass binning, this could be extremely useful
for electromagnetic follow-up of GW candidates Lynch  _et al._ (2018). Since
many such follow-ups are often target-of-opportunity observations, improving
the reliability of trigger information can be valuable in evaluating the
proper usage of scarce telescope time. The relative scarcity of noise triggers
could be a bigger computational issue in low latency, as the joint noise
distributions will have to be continuously reevaluated at a reasonable cadence
in order to account for the changing detector state Buikema  _et al._ (2020);
Acernese  _et al._ (2022). Therefore, further work would be needed to identify
methods that could potentially be used to reconstruct the noise distribution
in low latency.

Finally, Bayesian frameworks have been developed to assess the probability of
multimessenger detections Ashton  _et al._ (2018); Bartos  _et al._ (2019);
Veske  _et al._ (2021); Piotrzkowski  _et al._ (2022). These calculate the
probabilities associated with candidates being background noise or
astrophysical signals from different sources or the same source. Our unified
$p_{\mathrm{astro}}$ framework naturally feeds into these calculations.
Furthermore, our framework would enable the extension of these calculations to
consider how a nondetection in one messenger impacts the probability that a
candidate in a counterpart messenger is real. Given the rich science that may
result from a multimessenger discovery Abbott  _et al._ (2017c, 2020a);
Margutti and Chornock (2021), this may be a valuable avenue of future
investigation.

The data used in this paper for the analysis of GWTC-2.1 triggers is available
as a Zenodo repository zen .

## Acknowledgements

We thank Will Farr, Vicky Kalogera and Surabhi Sachdev for useful discussions.
We also thank Anarya Ray, Tom Dent and the anonymous referees for helpful
comments on the manuscript. S.B. was supported by National Science Foundation
(NSF) Grant No. PHY-2207945. C.P.L.B. acknowledges support from Science and
Technology Facilities Council (STFC) Grant No. ST/V005634/1. Z.D. acknowledges
support from the CIERA Board of Visitors Research Professorship. L.T. is
supported by the NSF through Grants No. OAC-2103662 and No. PHY-2011865. GSCD
acknowledges the STFC for funding through Grants No. ST/T000333/1 and No.
ST/V005715/1. The authors are grateful for computational resources provided by
the LIGO Laboratory and supported by NSF Grants No. PHY-0757058 and No.
PHY-0823459. This material is based upon work supported by NSF’s LIGO
Laboratory which is a major facility fully funded by the National Science
Foundation. This research has made use of data obtained from the Gravitational
Wave Open Science Center (gwosc.org) Abbott  _et al._ (2023), a service of
LIGO Laboratory, the LIGO Scientific Collaboration, the Virgo Collaboration,
and KAGRA. LIGO Laboratory and Advanced LIGO Aasi  _et al._ (2015) are funded
by the United States NSF as well as the STFC of the United Kingdom, the Max-
Planck-Society (MPS), and the State of Niedersachsen/Germany for support of
the construction of Advanced LIGO and construction and operation of the GEO
600 detector Dooley  _et al._ (2016). Additional support for Advanced LIGO was
provided by the Australian Research Council. Virgo Acernese  _et al._ (2015)
is funded, through the European Gravitational Observatory (EGO), by the French
Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions
by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, Spain. KAGRA Akutsu  _et al._ (2021) is supported by
Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan
Society for the Promotion of Science (JSPS) in Japan; National Research
Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia
Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan.
Corner plots were made with the Chainconsumer Hinton (2016) package. This
document carries the LIGO DCC number P2300107.

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  *[LIGO]: Laser Interferometer Gravitational-Wave Observatory
  *[GW]: gravitational wave
  *[FAR]: false alarm rate
  *[GWs]: gravitational wave
  *[LVK]: LIGO-Virgo-KAGRA
  *[O3]: third observing run
  *[CBC]: compact binary coalescence
  *[BBH]: binary black hole
  *[BNS]: binary neutron star
  *[KDE]: kernel density estimation
  *[KDEs]: kernel density estimation
  *[FARs]: false alarm rate
  *[BBHs]: binary black hole
  *[NSBH]: neutron star–black hole binary