An Order Statistics PostMortem on LIGOVirgo GWTC2 Events Analyzed with Nested Sampling_2

2025-04-27 0 0 770.96KB 8 页 10玖币

侵权投诉

RESEARCH ARTICLE

www.ann-phys.org

An Order Statistics Post-Mortem on LIGO–Virgo GWTC-2

Events Analyzed with Nested Sampling

Talya Klinger* and Michalis Agathos

The data analysis carried out by the LIGO–Virgo collaboration on

gravitational-wave events utilizes nested sampling to compute Bayesian

evidences and posterior distributions for inferring the source properties of

compact binaries. With poor sampling from the constrained prior, nested

sampling algorithms may misbehave and fail to sample the posterior

distribution faithfully. Fowlie et al. (2020) outlines a method of validating the

performance of nested sampling, or identifying pathologies such as plateaus

in the parameter space, using likelihood insertion order statistics. Here, this

method is applied to nested sampling analyses of all events in the ﬁrst and

second gravitational wave transient catalogs (GWTC-1 and GWTC-2) of the

LIGO–Virgo collaboration. The insertion order statistics are tested for

uniformity across 45 events in the catalog and it is found that, with a few

exceptions that have negligible eﬀect on the ﬁnal posteriors, the data from the

analysis of events in the catalog is consistent with unbiased prior sampling.

There is, however, weak evidence against uniformity at the catalog-level

meta-test, yielding a Kolmogorov–Smirnov meta-p-value of 1.44 ×10−3.

1. Introduction

Since the ﬁrst direct detection of gravitational waves (GWs) in

2015,[1] the LIGO–Virgo collaboration (LVC) has published the

detection of tens of GW signals emitted by coalescing black-

hole and neutron-star binaries, in the three observing runs car-

ried out so far (O1, O2, and O3).[2–4] In gravitational-wave data

analysis, parameter estimation is the process of inferring the

source properties of signals which have already been identiﬁed as

T. Klinger

Cardiﬀ University School of Physics and Astronomy

5 The Parade, Newport Road, Cardiﬀ CF24 3AA, UK

E-mail: talyaklinger@gmail.com

M. Agathos

DAMTP

Centre for Mathematical Sciences

University of Cambridge

Wilberforce Road, Cambridge CB3 0WA, UK

M. Agathos

Kavli Institute for Cosmology Cambridge

Madingley Road, Cambridge CB3 0HA, UK

The ORCID identiﬁcation number(s) for the author(s) of this article

can be found under https://doi.org/10.1002/andp.202200271

GmbH. This is an open access article under the terms of the Creative

Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/andp.202200271

gravitational waves produced by com-

pact binaries. Bayesian inference meth-

ods are employed to ﬁt waveform models

to data, using algorithms designed for ef-

ﬁciently sampling high-dimensional pa-

rameter spaces. One such algorithm is

nested sampling, a method for eﬃciently

computing Bayesian evidences as well as

posterior probability distributions, intro-

duced by Skilling in 2006[5] (for a review,

see ref. [6]). Here, we use a new method

of statistically verifying nested sampling

output, the insertion order cross-check

developed by Fowlie et al.,[7] to test for

biased nested sampling in the LVC’s

gravitational-wave data analysis.

The LVC uses nested sampling along-

side Markov Chain Monte Carlo (MCMC)

and occasionally RIFT[8] to obtain pos-

terior distributions in parameter esti-

mation. These algorithms serve nec-

essary and complementary purposes.

While the LVC’s implementation of MCMC converges faster

for signals with long inspiral times, and RIFT allows for di-

rect comparison to numerical relativity, they do not directly

compute evidence, delivering only the normalized posterior

and requiring further statistical calculations to estimate the ev-

idence, introducing signiﬁcant statistical errors. Nested sam-

pling computes the evidence directly, allowing for greater

accuracy.

In the LSC algorithm library (LAL), nested sampling was orig-

inally implemented in the LALInference package.[9,10] During the

third observing run (O3), the LVC (now LVK) Collaboration grad-

ually shifted its main data analysis pipelines to a newer Bayesian

inference library bilby,[11] a Python-based modular code which

combines LAL’s libraries for data infrastructure and waveform

modeling with third-party nested samplers. The insertion order

cross-check deﬁned later in this paper has already been imple-

mented in many bilby samplers, including CPNest,[12] nessai,[13]

and UltraNest.[14] However, the ﬁrst two observing runs, O1 and

O2, and the ﬁrst half of the third observing run, O3a, were ana-

lyzed using LALInference alone. In this work, we utilize the inser-

tion order cross-check to perform a post-mortem analysis on all

nested sampling output for GW events in O1, O2, and O3a, eval-

uating the validity of parameter estimation results for the LVC’s

event catalogs GWTC-1 and GWTC-2.[15]

This article is structured as follows. In Section 2 we brieﬂy

describe the nested sampling algorithm and the insertion order

statistics we use to estimate the validity of the LVC analyses. We

then describe our implementation for the parameter-estimation

www.advancedsciencenews.com www.ann-phys.org

dataset output by LALInference in Section 4 and present the re-

sults on the GW events in Section 5. Concluding remarks are

given in Section 6.

2. Insertion Order Statistics in Nested Sampling

2.1. Nested Sampling

For a given GW event associated with the coalescence of a com-

pact binary, we can describe its source properties by a parame-

ter vector 𝜃∈Θ,whereΘdenotes the corresponding parameter

space, including the mass and spin of each component, the dis-

tance to the source, its sky-location and orientation angles, time,

and phase of coalescence, as well as any additional parameters

relating to matter properties in case of a neutron star, orbital ec-

centricity, etc. Given the observed data D, our aim is to infer the

parameters 𝜃of the source, i.e. estimate the posterior distribu-

tion P(𝜃|D, ) under the assumption that our background infor-

mation about the nature of the source, the behavior of our de-

tectors and the validity of GR as the underlying theory is correct.

In Bayesian statistics, this amounts to updating our prior ex-

pectations quantiﬁed by P(𝜃|) by making appropriate use of

Bayes’ theorem

P(D|𝜃,)×P(𝜃|)=P(D|)×P(𝜃|D, )(1)

L(𝜃)×𝜋(𝜃)d𝜃=Z×p(𝜃)d𝜃

L(𝜃)=P(D|𝜃,), known as the likelihood function, and 𝜋(𝜃)=

P(𝜃|), the prior, give the desired quantities Z=P(D|), the evi-

dence and p=P(𝜃|D, ), the posterior. Computing the likelihood

function (the probability density for observing data D, given the

model and the true values of the parameters) requires models

for both the detector signal and noise—in LIGO’s case, LALSim-

ulation can generate a waveform model for the signal, while the

noise for each detector is assumed to be Gaussian and is charac-

terized by a power spectral density (PSD) which is pre-estimated

based on a stretch of data around the time of the event.[15] Infor-

mation from all detectors in operation is combined into a coher-

ent network likelihood which is the product of individual detec-

tor likelihoods.[10] The task of eﬃciently sampling the parameter

space to map the likelihood function, is carried out by the nested

sampling algorithm.

The evidence Z—the probability of observing the measured

data, given the model—is deﬁned as

Z=∫L(𝜃)𝜋(𝜃)d𝜃(2)

This is an important quantity in Bayesian data analysis, as

the evidences produced by diﬀerent models can be directly com-

pared. Hence, the evidence can be used to rank competing hy-

potheses and quantify how much a given model is supported by

the data. dX=𝜋(𝜃)d𝜃is known as the element of prior mass. If

the prior mass contained by a likelihood contour

X(𝜆)=∫L(𝜃)>𝜆

𝜋(𝜃)d𝜃(3)

is known, the evidence can be written as a 1D integral,

Z=∫1

L(X)dX(4)

which is more computationally manageable than integrating

across a high-dimensional parameter space Θ.

Nested sampling is a method for computing evidence that

takes advantage of this formulation, relying on the statistical

properties of prior sampling to provide a fast and accurate es-

timate of the prior mass at each integration step.

2.2. Summary of the Nested Sampling Algorithm

Nested sampling relies on sampling from the constrained prior:

points from the prior with likelihood higher than some mini-

mum value. As points from the constrained prior are sampled

and discarded throughout the algorithm, the samples used at

each step are called live points.

The nested sampling algorithm proceeds as follows:

1. Choose the number of live points nlive and sample nlive ini-

tial points from the constrained prior. Also, set an evidence

threshold 𝜖.

2. Identify the live point with the lowest likelihood L∗

i. Discard

the live point and record its likelihood.

3. Sample a new live point from 𝜋(𝜃) with L>L∗

i. At this stage,

the prior volume compresses exponentially, giving prior vol-

ume Xi≈exp(−1∕nlive) on the ith step (the proof is nontrivial,

see ref. [5]).

4. Integrate the evidence Ziusing L∗

iand Xi.

5. Repeat steps (2)–(4) until a stopping condition is reached:

LmaxXi∕Zi<e𝜖,whereLmax is the highest likelihood discov-

ered so far, Xiis the prior volume inside the current iso-

likelihood contour L∗

i,andZiis the current estimate of the

evidence. For LALInference,𝜖=0.1; essentially, if all the live

points were to have the maximum discovered likelihood, the

evidence would only change by a factor of less than 0.1.[10]

Nested sampling requires faithful sampling from the con-

strained prior to produce accurate evidences and posteriors. In

practice, sampling from the entire prior and accepting only

points with high enough likelihood is impractically slow, be-

cause the volume of acceptable points decreases exponentially

in time. So, most implementations of nested sampling sample

from a restricted region of parameter space drawn around the

live points. LALInference, in particular, generates samples by an

MCMC chain from a randomly chosen previous livepoint, and

choosing the length of the MCMC chain is a tradeoﬀ between

speed and accuracy.[10]

If the restricted region is too small or the MCMC chains too

short, the constrained prior may not fully cover the iso-likelihood

contour, violating the fundamental assumptions of nested sam-

pling. Plateaus—regions of constant L(𝜃)—also violate the as-

sumptions of nested sampling, causing live points to be nonuni-

formly distributed in X.

2.3. Insertion Order Crosscheck

The insertion index is the position where an element must be

inserted in a sorted list to preserve order. More concretely, if xis

a sorted list and there exists a sample ysuch that

xi−1<y<xi(5)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

RESEARCHARTICLEwww.ann-phys.orgAnOrderStatisticsPost-MortemonLIGO–VirgoGWTC-2EventsAnalyzedwithNestedSamplingTalyaKlinger*andMichalisAgathosThedataanalysiscarriedoutbytheLIGO–Virgocollaborationongravitational-waveeventsutilizesnestedsamplingtocomputeBayesianevidencesandposteriordistributionsforinfer...

展开>> 收起<<

An Order Statistics PostMortem on LIGOVirgo GWTC2 Events Analyzed with Nested Sampling_2.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

An Order Statistics PostMortem on LIGOVirgo GWTC2 Events Analyzed with Nested Sampling_2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: