# Use of the MultiNest algorithm for gravitational wave data analysis

**ABSTRACT** We describe an application of the MultiNest algorithm to gravitational wave data analysis. MultiNest is a multimodal nested sampling algorithm designed to efficiently evaluate the Bayesian evidence and return posterior probability densities for likelihood surfaces containing multiple secondary modes. The algorithm employs a set of live points which are updated by partitioning the set into multiple overlapping ellipsoids and sampling uniformly from within them. This set of live points climbs up the likelihood surface through nested iso-likelihood contours and the evidence and posterior distributions can be recovered from the point set evolution. The algorithm is model-independent in the sense that the specific problem being tackled enters only through the likelihood computation, and does not change how the live point set is updated. In this paper, we consider the use of the algorithm for gravitational wave data analysis by searching a simulated LISA data set containing two non-spinning supermassive black hole binary signals. The algorithm is able to rapidly identify all the modes of the solution and recover the true parameters of the sources to high precision. Comment: 18 pages, 4 figures, submitted to Class. Quantum Grav; v2 includes various changes in light of referee's comments

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**ABSTRACT:**We consider the concept of fundamental bias in gravitational wave astrophysics as the assumption that general relativity is the correct theory of gravity during the entire wave-generation and propagation regime. Such an assumption is valid in the weak field, as verified by precision experiments and observations, but it need not hold in the dynamical strong-field regime where tests are lacking. Fundamental bias can cause systematic errors in the detection and parameter estimation of signals, which can lead to a mischaracterization of the Universe through incorrect inferences about source event rates and populations. We propose a remedy through the introduction of the parametrized post-Einsteinian framework, which consists of the enhancement of waveform templates via the inclusion of post-Einsteinian parameters. These parameters would ostensibly be designed to interpolate between templates constructed in general relativity and well-motivated alternative theories of gravity, and also include extrapolations that follow sound theoretical principles, such as consistency with conservation laws and symmetries. As an example, we construct parametrized post-Einsteinian templates for the binary coalescence of equal-mass, nonspinning compact objects in a quasicircular inspiral. The parametrized post-Einsteinian framework should allow matched filtered data to select a specific set of post-Einsteinian parameters without a priori assuming the validity of the former, thus either verifying general relativity or pointing to possible dynamical strong-field deviations.Physical review D: Particles and fields 12/2009; 80(12):122003-122003. -
##### Article: Improving Bayesian analysis for LISA Pathfinder using an efficient Markov Chain Monte Carlo method

Luigi Ferraioli, Edward K. Porter, Michele Armano, Heather Audley, Giuseppe Congedo, Ingo Diepholz, Ferran Gibert, Martin Hewitson, Mauro Hueller, Nikolaos Karnesis, Natalia Korsakova, Miquel Nofrarias, Eric Plagnol, Stefano Vitale[Show abstract] [Hide abstract]

**ABSTRACT:**We present a parameter estimation procedure based on a Bayesian framework by applying a Markov Chain Monte Carlo algorithm to the calibration of the dynamical parameters of the LISA Pathfinder satellite. The method is based on the Metropolis-Hastings algorithm and a two-stage annealing treatment in order to ensure an effective exploration of the parameter space at the beginning of the chain. We compare two versions of the algorithm with an application to a LISA Pathfinder data analysis problem. The two algorithms share the same heating strategy but with one moving in coordinate directions using proposals from a multivariate Gaussian distribution, while the other uses the natural logarithm of some parameters and proposes jumps in the eigen-space of the Fisher Information matrix. The algorithm proposing jumps in the eigen-space of the Fisher Information matrix demonstrates a higher acceptance rate and a slightly better convergence towards the equilibrium parameter distributions in the application to LISA Pathfinder data. For this experiment, we return parameter values that are all within ∼1σ of the injected values. When we analyse the accuracy of our parameter estimation in terms of the effect they have on the force-per-unit of mass noise, we find that the induced errors are three orders of magnitude less than the expected experimental uncertainty in the power spectral density.Experimental Astronomy 02/2014; · 2.97 Impact Factor - SourceAvailable from: export.arxiv.org
##### Article: Gravitational Wave Tests of General Relativity with Ground-Based Detectors and Pulsar Timing Arrays

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**ABSTRACT:**This review is focused on tests of Einstein's theory of General Relativity with gravitational waves that are detectable by ground-based interferometers and pulsar timing experiments. Einstein's theory has been greatly constrained in the quasi-linear, quasi-stationary regime, where gravity is weak and velocities are small. Gravitational waves will allow us to probe a complimentary, yet previously unexplored regime: the non-linear and dynamical strong-field regime. Such a regime is, for example, applicable to compact binaries coalescing, where characteristic velocities can reach fifty percent the speed of light and compactnesses can reach a half. This review begins with the theoretical basis and the predicted gravitational wave observables of modified gravity theories. The review continues with a brief description of the detectors, including both gravitational wave interferometers and pulsar timing arrays, leading to a discussion of the data analysis formalism that is applicable for such tests. The review ends with a discussion of gravitational wave tests for compact binary systems.04/2013;

Page 1

Use of the MULTINEST algorithm for gravitational wave

data analysis

Farhan Feroz1, Jonathan R. Gair2, Michael P. Hobson1and Edward

K. Porter3

1Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK

2Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK

3APC, UMR 7164, Universit´ e Paris 7 Denis Diderot, 10, rue Alice Domon et L´ eonie Duquet,

75205 Paris Cedex 13, France

Abstract.

data analysis. MULTINEST is a multimodal nested sampling algorithm designed to efficiently

evaluate the Bayesian evidence and return posterior probability densities for likelihood

surfaces containing multiple secondary modes. The algorithm employs a set of ‘live’ points

which are updated by partitioning the set into multiple overlapping ellipsoids and sampling

uniformly from within them. This set of ‘live’ points climbs up the likelihood surface through

nested iso-likelihood contours and the evidence and posterior distributions can be recovered

from the point set evolution. The algorithm is model-independent in the sense that the specific

problem being tackled enters only through the likelihood computation, and does not change

how the ‘live’ point set is updated. In this paper, we consider the use of the algorithm for

gravitational wave data analysis by searching a simulated LISA data set containing two non-

spinning supermassive black hole binary signals. The algorithm is able to rapidly identify all

the modes of the solution and recover the true parameters of the sources to high precision.

We describe an application of the MULTINEST algorithm to gravitational wave

PACS numbers: 04.25.Nx, 04.30.Db, 04.80.Cc

arXiv:0904.1544v2 [gr-qc] 23 Jul 2009

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MULTINEST for gravitational wave data analysis2

1. Introduction

There is currently much active research into data analysis algorithms for gravitational wave

detectors, both on the ground and for the proposed space-based gravitational wave detector,

the Laser Interferometer Space Antenna (LISA) [1]. Research into LISA data analysis is

being encouraged by the Mock LISA Data Challenges [2] (MLDC), which so far have

included data sets containing individual and multiple white-dwarf binaries, a realisation of the

whole galaxy of compact binaries, single and multiple non-spinning supermassive black hole

(SMBH) binaries, either isolated or on top of a galactic confusion background and isolated

extreme-mass-ratio inspiral (EMRI) sources in purely instrumental noise. While most of

these Challenges have been solved in the sense that several groups returned solutions that

matched the injected parameters, the EMRI case caused particular difficulty [3, 4, 5, 6]. These

problems arose primarily because the likelihood surface contains many secondary maxima,

and so algorithms such as Markov Chain Monte Carlo (MCMC) tended to become stuck

on secondary maxima and were unable to find the primary mode. In Challenge 3 of the

MLDC,twonewsourceswereintroducedthathavesimilarmulti-modallikelihoodsurfaces—

spinning black hole binaries and cosmic string cusps [7]. The final analysis of the LISA data

will also have to contend with the presence of multiple overlapping sources simultaneously

present in the data stream. For these reasons, it is desirable to have algorithms that are able

to simultaneously identify and characterize all the modes of the likelihood surface. One

approach is to use an evolutionary algorithm, and a recent implementation of these ideas

is described in [8]. However, there are also existing tools that have been developed for other

applications that have the necessary features to be useful for the gravitational wave case. One

such algorithm is MULTINEST [9, 10], which is a multi-modal nested sampling algorithm.

The nested sampling algorithm [11] was developed as a tool for evaluating the Bayesian

evidence. It employs a set of ‘live’ points, each of which represents a particular set of

parameters in the multi-dimensional search space.

progresses, and climb together through nested contours of increasing likelihood. At each step

the algorithm works by finding a point of higher likelihood than the lowest likelihood point in

the ‘live’ point set and then replacing the lowest likelihood point with the new point. Nested

sampling has been applied to the issue of model selection for ground based observations of

gravitational waves [12] and forms part of the evolutionary algorithm for LISA data analysis

discussed in [8]. However, the primary difficulty in using nested sampling is to efficiently

sample points of higher likelihood to allow the points to climb up the likelihood surface.

MULTINEST solves the problem of efficient sampling by using the current set of ‘live’

points as a model of the shape of the likelihood surface. The idea is to decompose the

set of ‘live’ points into a sequence of overlapping ellipsoids, and then sample from within

one of these ellipsoids which is chosen at random from the current set. The algorithm is

model-free in the sense that the evolution of the ‘live’ point set does not make use of any

knowledge about the properties of the likelihood surface. The only model-specific element

is the subroutine which computes the likelihood at a given point in parameter space. This

approach reduces much of the computational overhead associated with evaluating Fisher

Matrices etc. in order to move the ‘live’ points on the likelihood surface. MULTINEST

has proven to be able to efficiently find and separate modes of a likelihood surface in many

different applications [13, 14, 15, 16]. This suggests it may be a powerful tool for gravitational

wave data analysis as well, and in this paper we explore these possibilities using a search for

non-spinning SMBH binaries as a test case. Such sources are known to possess a degeneracy

in the likelihood, since at low frequency the response of the detector to the true sky position

and the point antipodal to it on the sky can not be distinguished. We can therefore use these

These points move as the algorithm

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MULTINEST for gravitational wave data analysis3

sources as a test case to investigate the ability of MULTINEST to explore a multi-modal

likelihood in the context of gravitational wave detection.

The coalescences of supermassive black holes (SMBHs) in binaries are expected to be

a major source of gravitational waves (GW) for LISA, and we should be able to detect such

systems out to a redshift of z ∼ 10 with intrinsic parameter errors of a fraction of a percent

to a few percent and extrinsic parameter errors of a few to a few tens of percent [17, 18].

The detection problem for non-spinning SMBH binaries has already been solved, in the sense

that several groups have been able to successfully detect and recover parameters for such

systems both in controlled studies [18, 19] and in blind tests in the context of the MLDC.

Several algorithms have been proven to be successful — Metropolis Hastings Monte Carlo

(MHMC) [18, 19], template-bank based searches [20, 21], time-frequency methods [22] and

hierarchical searches mixing time-frequency and MHMC techniques [23]. More recently,

the evolutionary algorithm has also been shown to be effective in searches for non-spinning

SMBH binaries [8]. Thus, although the application of MULTINEST to these systems will be

an interesting test case, it is not essential to the success of LISA. However, the real power of

the algorithm will be in its application to multi-modal likelihood surfaces for systems such

as EMRIs. The present work is an indication of the power of the technique and a stimulus

for further research on this algorithm in the context of gravitational wave data analysis. In

addition, the evidence values returned by MULTINEST can be used for model selection,

which will also be important for LISA. Bayesian model selection using MCMC techniques

has already been explored in the LISA context [24] for the case of white-dwarf binaries.

MULTINEST provides an alternative approach to addressing similar questions, although we

will not discuss that application of the algorithm in the present paper.

The paper is organised as follows. In section 2 we give a brief introduction to Bayesian

inference and then give details of the MULTINEST algorithm in section 3, including an

overviewofnestedsamplingandadescriptionoftheellipsoidalsamplingscheme. Insection4

we briefly describe the waveform and noise models used in this analysis. In section 5 we

describe an application of the algorithm to a simultaneous search for two non-spinning black

hole binaries in a single data set. We provide results on the parameter recovery achieved by

the algorithm and compare these to the one sigma error estimates from the Fisher Matrix. We

finish in section 6 with a summary and a discussion of possible future applications of this

work.

2. Bayesian Inference

Our detection methodology is built upon the principles of Bayesian inference, and so we

begin by giving a brief summary of this framework. Bayesian inference methods provide a

consistent approach to the estimation of a set of parameters Θ in a model (or hypothesis) H

for the data D. Bayes’ theorem states that

Pr(Θ|D,H) =Pr(D|Θ,H)Pr(Θ|H)

Pr(D|H)

where Pr(Θ|D,H) ≡ P(Θ) is the posterior probability distribution of the parameters,

Pr(D|Θ,H) ≡ L(Θ) is the likelihood, Pr(Θ|H) ≡ π(Θ) is the prior distribution, and

Pr(D|H) ≡ Z is the Bayesian evidence.

Bayesian evidence is simply the factor required to normalise the posterior over Θ and is

given by:

?

,

(1)

Z =

L(Θ)π(Θ)dNΘ,

(2)

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MULTINEST for gravitational wave data analysis4

where N is the dimensionality of the parameter space. Since the Bayesian evidence is

independent of the parameter values Θ, it is usually ignored in parameter estimation problems

and the posterior inferences are obtained by exploring the un–normalized posterior using

standard MCMC sampling methods.

Bayesian parameter estimation has been used quite extensively in a variety of

astronomical applications, including gravitational wave astronomy, although standard MCMC

methods, such as the basic Metropolis–Hastings algorithm or the Hamiltonian sampling

technique (see e.g. [27]), can experience problems in sampling efficiently from a multi–

modal posterior distribution or one with large (curving) degeneracies between parameters.

Moreover, MCMC methods often require careful tuning of the proposal distribution to sample

efficiently, and testing for convergence can be problematic.

In order to select between two models H0and H1one needs to compare their respective

posterior probabilities given the observed data set D, as follows:

Pr(H1|D)

Pr(H0|D)=Pr(D|H1)Pr(H1)

Pr(D|H0)Pr(H0)=Z1

Z0

Pr(H1)

Pr(H0),

(3)

where Pr(H1)/Pr(H0) is the prior probability ratio for the two models, which can often

be set to unity but occasionally requires further consideration (see, for example, [13, 14] for

the cases where the prior probability ratios should not be set to unity). It can be seen from

Eq. (3) that the Bayesian evidence plays a central role in Bayesian model selection. As the

average of likelihood over the prior, the evidence automatically implements Occam’s razor:

a simpler theory which agrees well enough with the empirical evidence is preferred. A more

complicated theory will only have a higher evidence if it is significantly better at explaining

the data than a simpler theory.

Unfortunately, evaluation of Bayesian evidence involves the multidimensional integral

(Eq. (2)) and thus presents a challenging numerical task.

thermodynamic integration [28, 24] are extremely computationally expensive which makes

evidence evaluation typically at least an order of magnitude more costly than parameter

estimation. Some fast approximate methods have been used for evidence evaluation, such

as treating the posterior as a multivariate Gaussian centred at its peak (see, for example,

[25]), but this approximation is clearly a poor one for highly non-Gaussian and multi–modal

posteriors. Various alternative information criteria for model selection are discussed in [26],

but the evidence remains the preferred method.

Standard techniques like

3. Nested Sampling and the MULTINEST Algorithm

Nested sampling [11] is a Monte Carlo method targetted at the efficient calculation of the

evidence, but also produces posterior inferences as a by-product. It calculates the evidence

by transforming the multi–dimensional evidence integral into a one–dimensional integral that

is easy to evaluate numerically. This is accomplished by defining the prior volume X as

dX = π(Θ)dDΘ, so that

?

where the integral extends over the region(s) of parameter space contained within the iso-

likelihood contour L(Θ) = λ. The evidence integral, Eq. (2), can then be written as

Z =

0

X(λ) =

L(Θ)>λ

π(Θ)dNΘ,

(4)

?1

L(X)dX,

(5)

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MULTINEST for gravitational wave data analysis5

(a)(b)

Figure 1. Cartoon illustrating (a) the posterior of a two dimensional problem; and (b) the

transformed L(X) function where the prior volumes Xiare associated with each likelihood

Li.

where L(X), the inverse of Eq. (4), is a monotonically decreasing function of X. Thus, if one

can evaluate the likelihoods Li= L(Xi), where Xiis a sequence of decreasing values,

0 < XM< ··· < X2< X1< X0= 1,

as shown schematically in Fig. 1, the evidence can be approximated numerically using

standard quadrature methods as a weighted sum

?

where the weights wifor the simple trapezium rule are given by wi=1

example of a posterior in two dimensions and its associated function L(X) is shown in Fig. 1.

(6)

Z =

M

i=1

Liwi,

(7)

2(Xi−1− Xi+1). An

3.1. Evidence Evaluation

In order to evaluate the evidence value given in (Eq. 7), a set of N ‘live’ points are drawn

uniformly from the full prior π(Θ) with the initial prior volume X0set to 1. The likelihood

value for each of the N ‘live’ points is calculated and the point with the minimum likelihood

value L0 is removed from the ‘live’ point set and is replaced by another point sampled

uniformly from the prior with likelihood L > L0. This results in the reduction in the prior

volume within the iso-likelihood contour by a factor t1i.e. X1= t1X0where t1follows the

distribution of the largest of the N samples drawn uniformly from the interval [0,1] which is

given by Pr(t) = NtN−1. This procedure is repeated at each subsequent iteration i, at which

the point with the lowest likelihood value Liis replaced by a new point sampled uniformly

from the prior with L > Liuntil the entire prior volume has been traversed. The expected

value and the standard deviation of logt, which dominates the geometrical exploration is

given by:

E[logt] = −1/N,

With the values of logt at different iterations being independent of each other, the prior

volume at iteration i is expected to be:

logXi≈ −(i ±

Thus, one takes

Xi= exp(−i/N).

σ[logt] = 1/N.

(8)

√i)/N.

(9)

(10)

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MULTINEST for gravitational wave data analysis6

3.2. Stopping Criterion

The primary aim of nested sampling algorithm is to calculate the evidence value and so it

should be terminated when the evidence value has been calculated to required accuracy. One

way to achieve this is by checking whether the change in the evidence value from one iteration

to the other is smaller than a specified tolerance but in cases where the posterior contains

narrow peaks close to its maximum, this can result in an underestimated evidence value.

Skilling [11] provides a robust stopping condition by calculating an upper limit on the

evidencethatcanbedeterminedfromthesetofcurrent‘live’points. Thislimitiscalculatedby

assuming that all the ‘live’ points have the likelihood value equal to the maximum–likelihood

Lmaxand so the largest evidence contribution that can be made by the remaining portion of

the posterior is ∆Zi = LmaxXi. Lmaxcan be taken to be the maximum likelihood value

amongst the ‘live’ points. At the beginning of the nested sampling algorithm, it is possible for

the high likelihood regions of the posterior to not have been explored adequately resulting in

the maximum–likelihood value Lmax, amongst the ‘live’ points being a lot smaller than the

true maximum–likelihood value. The remaining prior volume Xion the other hand would

be quite high and therefore the largest evidence contribution ∆Zi = LmaxXi would be

high as well allowing the algorithm to proceed. At subsequent stages, the remaining prior

volume Xidecreases exponentially as given by (Eq. 10) and the high likelihood regions of

the posteriors are explored in greater detail with estimated Lmaxgetting closer and closer to

the true maximum–likelihood value. The largest evidence contribution ∆Zi= LmaxXi, thus

provides a robust stopping criteria.

We choose to stop when this quantity would no longer change the final evidence estimate

by 0.5 in log–evidence.

3.3. Posterior Inferences

Although the nested sampling algorithm has been designed to calculate the evidence value,

accurate posterior inferences can also be generated as a by-product. Once the evidence Z is

found, each one of the final ‘live’ points and the discarded points from the nested sampling

process, i.e., the points with the lowest likelihood value at each iteration i of the algorithm is

simply assigned the probability weight

pi=Liwi

Z

These samples can then be used to calculate inferences of posterior parameters such as

means, standard deviations, covariances and so on, or to construct marginalised posterior

distributions.

.

(11)

3.4. MULTINEST Algorithm

The most challenging task in implementing the nested sampling algorithm is drawing samples

from the prior within the hard constraint L > Li at each iteration i.

naive approach that draws blindly from the prior would result in a steady decrease in the

acceptance rate of new samples with decreasing prior volume (and increasing likelihood). The

MULTINEST algorithm [9, 10] tackles this problem through an ellipsoidal rejection sampling

scheme by enclosing the ‘live’ point set within a set of (possibly overlapping) ellipsoids

and a new point is then drawn uniformly from the region enclosed by these ellipsoids. The

number of points in an individual ellipsoid and the total number of ellipsoids is decided by

an ‘expectation–maximization’ algorithm so that the total sampling volume, which is equal

Employing a

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MULTINEST for gravitational wave data analysis7

(a)(b)

Figure 2. Illustrations of the ellipsoidal decompositions performed by MULTINEST. The

points given as input are overlaid on the resulting ellipsoids. 1000 points were sampled

uniformly from: (a) two non-intersecting ellipsoids; and (b) a torus.

to the sum of volumes of the ellipsoids, is minimized. This allows maximum flexibility and

efficiency by breaking up a mode resembling a Gaussian into a relatively small number of

ellipsoids, and if the posterior mode possesses a pronounced curving degeneracy so that it

more closely resembles a (multi–dimensional) ‘banana’ then it is broken into a relatively

large number of small ‘overlapping’ ellipsoids (see Fig. 2). With enough ‘live’ points, this

approach allows the detection of all the modes simultaneously, resulting in typically two

orders of magnitude improvement in efficiency and accuracy over standard methods for

inference problems in cosmology and particle physics phenomenology (see, for example,

[13, 14, 15, 16]).

The ellipsoidal decomposition scheme described above also provides a mechanism for

mode identification. By forming chains of overlapping ellipsoids (enclosing the ‘live’ points),

the algorithm can identify distinct modes with distinct ellipsoidal chains, e.g., in Fig. 2 panel

(a) the algorithm identifies two distinct modes while in panel (b) the algorithm identifies only

one mode as all the ellipsoids are linked with each other because of the overlap between them.

Once distinct modes have been identifed, they are evolved independently.

Another feature of the MULTINEST algorithm is the evaluation of the global as well as

the ‘local’ evidence values associated with each mode. These evidence values can be used

in calculating the probability that an identified ‘local’ peak in the posterior corresponds to a

real object (see, for instance, [12, 13, 14]). We defer the discussion of quantifying the SMBH

detection to a later work.

4. Problem Description

4.1. Waveform model

The waveform from a binary composed of two non-spinning black holes depends on nine

parameters:?λ = {ln(Mc),ln(µ),θ,φ,ln(tc),ι,ϕc,ln(DL),ψ}, where Mcis the chirp mass,

µ is the reduced-mass, (θ,φ) are the sky location of the source, tcis the time-to-coalescence,

ι is the inclination of the orbit of the binary, ϕcis the phase of the GW at coalescence, DL

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MULTINEST for gravitational wave data analysis8

is the luminosity distance and ψ is the polarization of the GW measured in the Solar System

barycentre (SSB). ψ is the angle between the principal polarisation axes of the source and the

“natural” polarisation axes to use in the SSB, which are perpendicular to the line of sight to

the source and the orbital angular momentum of the binary. We model the waveform using the

restricted post-Newtonian approximation, in which the two polarizations of the GW are [29]

?1 + cos2ι??Gmω

h×= −4Gmη

c2DL

c3

where m = m1+ m2is the total mass of the binary, η = m1m2/m2is the reduced mass

ratio, G is Newton’s constant and c is the speed of light. The chirp mass and reduced mass

are given in terms of m and η as Mc= mη3/5and µ = mη. The function ω = dΦorb/dt is

the orbital frequency, and Φ = ϕc− ϕ(t) = 2Φorbis the gravitational wave phase. We take

these to 2PN order

c3

8Gm

?1855099

Φ(t) = ϕc−2

η

?9275495

where

c3η

5Gm(tc− t).

The above model describes the inspiral only and not the merger or ringdown phase. To avoid

artifacts when taking Fourier transforms, we follow the approach introduced in [18] and now

used in the MLDC [2] and continue the inspiral until the orbit reaches the innermost stable

circular orbit at 6M, but employ a hyperbolic taper from 7M to ensure the waveform goes

smoothly to zero as the end of the inspiral approaches. The taper has the value of 1 for

r > 7M and then smoothly goes to zero at r = 6M, where the waveform finishes. To

implement the LISA response, we use the low-frequency approximation, as described in [30].

In the low-frequency limit we expect there to be bright modes in the likelihood at both the

true sky position and at a position approximately antipodal to it, as mentioned earlier. The

antipodal position corresponds to the shift θ → π − θ, φ → φ + π. The ‘antipodal’ mode

of the likelihood is strictly antipodal at low frequencies, but moves as the gravitational wave

frequency increases.

The four parameters {DL,ι,ϕc,ψ} are extrinsic parameters which only affect how the

gravitational waveform phase at the detector projects into a detector response. It is possible to

search over these automatically using a generalisation of the F-statistic [31]. Details of how

the F-statistic is computed for non-spinning SMBH binaries may be found in [18]. We make

use of the F-statistic maximization in the first stage of our search.

A gravitational wave search is sensitive to redshifted masses, ¯ m1= (1 + z)m1, rather

than the intrinsic masses. All our subsequent results will be quoted for redshifted mass

quantities, which we will denote with an overbar for clarity.

h+=2Gmη

c2DL

c3

?2

3

cos(Φ),

(12)

cosι

?Gmω

?2

3

sin(Φ),

(13)

ω(t) =

?

Θ−3/8+

?743

258048η +371

?3715

258048η +1855

2688+1132η

?

Θ−5/8−3π

?

Θ3/8−3π

?

10Θ−3/4

?

4Θ1/4

?

+

14450688+56975

?

14450688+284875

2048η2

?

2048η2

Θ−7/8

(14)

Θ5/8+

8064+5596η

+

Θ1/8

,

(15)

Θ(t;tc) =

(16)

Page 9

MULTINEST for gravitational wave data analysis9

4.2. Likelihood evaluation

The gravitational waveform signals can be thought of as occupying a vector space, on which

there is a natural scalar product [32, 33]

?∞

where

?∞

is the Fourier transform of the time domain waveform h(t). The quantity Sn(f) is the one-

sided noise spectral density of the detector. For a family of sources with waveforms h(t;?λ)

that depend on parameters?λ, the output of the detector, s(t) = h(t;?λ0) + n(t), consists of

the true signal h(t;?λ0) and a particular realisation of the noise, n(t). Assuming that the noise

is stationary and Gaussian, the logarithm of the likelihood that the parameter values are given

by?λ is

??λ

and it is this log-likelihood that is evaluated at each ‘live’ point in the search. The constant,

C, depends on the dimensionality of the search space, but its value is not important as we are

only interested in the relative likelihoods of different points.

In section 5 we will compare the errors in our recovered parameters to the theoretical

noise-induced errors. The latter can be computed as σi=?(Γ−1)iiwhere

Γij=

∂λi

∂λj

is the Fisher Information Matrix (FIM).

?h|s? = 2

0

df

Sn(f)

?˜h(f)˜ s∗(f) +˜h∗(f)˜ s(f)

?

.

(17)

˜h(f) =

−∞

dth(t)e2πıft

(18)

logL

?

= C −

?

s − h

??λ

????s − h

??λ

??

/2,

(19)

?∂h

????

∂h

?

(20)

4.3. Noise model

The noise consists of an instrumental part, which we take from [34]

1

4L2

?

where L = 5 × 106km is the arm-length for LISA, Spos

Sacc

has units of Hz−1. The quantity f∗= 1/(2πL) is the mean transfer frequency for the LISA

arms. In addition, there is a noise contribution from the confusion foreground of galactic

compact binaries. We represent this using the model of [35, 36]

Sinst

n

(f) =

?

2Spos

n (f)

?

2 +

?f

?f

f∗

2??

???

+ 8Sacc

n(f) 1 + cos2

f∗

1

(2πf)4+(2π10−4Hz)2

(2πf)6

??

.

(21)

n (f) = 4 × 10−22m2/Hz and

n(f) = 9×10−30m2/s4/Hz are the position and acceleration noise respectively. Sinst

n

(f)

Sconf

n

(f) =

10−44.62(f/Hz)−2.3

10−4< (f/Hz) ≤ 10−3

10−3< (f/Hz) ≤ 10−2.7

10−2.7< (f/Hz) ≤ 10−2.4

10−2.4< (f/Hz) ≤ 10−2

10−50.92(f/Hz)−4.4

10−62.8(f/Hz)−8.8

10−89.68(f/Hz)−20

Hz−1,

(22)

Page 10

MULTINEST for gravitational wave data analysis 10

m1/M?

1 × 107

4 × 106

m2/M?

1 × 106

1 × 106

tc/yrsθφzιψϕc

1

2

0.90

1.02

0.6283

2.1206

4.7124

3.9429

1.0

0.8

1.1120

0.6565

1.2330

1.0646

2.2220

1.5116

Table 1. Parameter values for the two SMBHBs considered in this study. The dual-TDI

channel SNRs for the sources are 200 and 131 respectively.

The total noise is the sum of these contributions. At low frequencies, the LISA detector can

be thought to consist of two independent right-angle interferometers with independent noise

described by the same power spectral density. The total likelihood is obtained by summing

the scalar product (17) over the two detectors, and the FIM of the network is similarly given

by the sum of the two FIMs.

5. Results of MULTINEST Search

To test the MULTINEST algorithm we generated a data set containing two SMBH binaries,

one of which coalesced during the observation time, and one of which coalesced a few days

after the end of the observation. We took these sources to have the same parameters as the

two sources searched for by the evolutionary algorithm in [8] to facilitate direct comparison

between the two techniques. The parameters of the two sources are summarised in Table 1.

We used a standard flat WMAP cosmology, with radiation, matter and dark energy densities

(ΩR,ΩM,ΩΛ) = (4.9 × 10−5,0.27,0.73) and H0=71 km/s/Mpc [37]. The redshifts of the

sources therefore correspond to luminosity distances of 6.634Gpc and 5.024Gpc respectively.

The redshifted chirp mass and reduced mass are (¯

for source 1, and (2.997 × 106,1.44 × 106)M?for source 2, and the full, dual-TDI channel,

signal-to-noise ratios (SNRs) are 200 and 131 respectively.

For the first stage of the search, we included the F-statistic maximization in the

likelihood evaluation and used MULTINEST with 1000 ‘live’ points to search only the five

dimensional space of intrinsic parameters ( ¯

m1, ¯

search of ¯

m1, ¯

In general, the number of ‘live’ points required for a search is problem specific, depending

on the dimensionality of the search space, and on the complexity of the likelihood surface.

Therefore, the ideal number must be found by trial and error. Using too few ‘live’ points leads

tounder-samplingoftheposterior. Itisnormallyclearfromtheposteriordistributionswhether

the likelihood has been properly sampled or not. Using more ‘live’ points than necessary does

not affect the results, but is computationally more expensive. The choice of 1000 in this

case was based on previous experience in other contexts, but appeared to work well. The

algorithm returned 11 possible modes and, based on the time of coalesence, it was clear that

five were associated with the coalescing source, and six with the non-coalescing source. Four

modes were the same secondaries found using the evolutionary algorithm [8], each of which

corresponded to mass and spin values close to those of one of the sources, but had sky position

in the vicinity of one of the two sky solutions of the other source. Of the three additional

secondaries identified in the likelihood, one was associated with the coalescing source, and

wasveryclosetothetruevaluesofthemassesandtc, butwasalsointhewrongpositiononthe

sky. The other two secondaries were associated with the non-coalescing source and had mass

and tcparameters that were relatively close to the true values, although many Fisher Matrix

Mc, ¯ µ) = (4.9289×106,1.8182×106)M?

m2,θ,φ,tc). We used wide priors for this

m2∈?5 × 105,3 × 107?M?, tc∈ [0.85,1.1]yrs, θ ∈ [0,π] and φ ∈ [0,2π].

Page 11

MULTINEST for gravitational wave data analysis11

σ away, but had incorrect sky locations. It is likely that all of these secondaries correspond

to parameter space points for which the waveforms match the brighter waveform cycles that

come toward the end of the inspiral, but not the earlier part of the waveform. This stage of the

search took approximately one and a half hours to run on a single 3.0 GHz Intel Woodcrest

processor, and used ∼ 170,000 likelihood evaluations.

In Table 2 we list the mean and standard deviation of the posteriors recovered by the

algorithm in this stage, for the two true and the two antipodal modes of the sources. In Table 3

we assess the error in parameter recovery as a multiple of the theoretical error, computed from

the Fisher matrix. This is one of the standard approaches used to assess entries to the LISA

Mock Data Challenge and so we include it for easier comparison to other algorithms discussed

in the literature. The posterior distributions are not Gaussian, which is assumed for the Fisher

matrix estimate, so we do not necessarily expect these to be an accurate indication of the

true error. In practice, we will estimate the uncertainty in the parameter estimation from

the recovered posterior and, in all cases, the true solution lies within 1-σ of the peak of the

recovered posterior, when σ is computed from the posterior in that way. We quote results in

Table 3 for the two true solutions, and for the two antipodal sky solutions. Errors are quoted

for the redshifted chirp mass, ¯

Mc, and reduced mass, ¯ µ, rather than ¯

former are used more conventionally in the literature and this therefore facilitates comparison

to other work, in particular the evolutionary algorithm [8].

We see that the algorithm recovered all of the intrinsic parameters to extremely high

precision, being within 1σ of the true parameters for all but the antipodal sky solution of the

second source. As mentioned earlier, the ‘antipodal’ solution is no longer exactly antipodal

on the sky at higher frequencies. Thus, the apparently larger errors in the parameters for the

antipodal solution do not mean that the algorithm failed to find the true peak of the likelihood,

but merely that the secondary is shifted from the precisely antipodal position. These results

can be directly compared to results obtained using the evolutionary algorithm in [8]. The

MULTINEST results are slightly better than those of the evolutionary algorithm, although in

both cases the results were within the error we would expect from noise fluctuations and so

this difference could easily be due to a difference in the noise realisation used to generate

the data. We note that the evolutionary algorithm also recovered parameters for the antipodal

solution of the non-coalescing source that were about 4σ from the true parameters, which is

consistent with the prior explanation for that difference. Presently, the MULTINEST algorithm

runs about five to ten times faster than the evolutionary algorithm, but the latter has not yet

been optimized and further work is needed to understand how the efficiency and speed of both

algorithms will scale when they are applied to other types of gravitational wave source.

The fact that the intrinsic parameters of the coalescing source are all so well determined

is due to a combination of the particular noise realisation used and the strong correlations

between the intrinsic parameters. The correlations ensure that if one of the parameters is well-

determined then all of the parameters will be well-determined. The results seem surprising

because the error is so low in all of the parameters, which, if the parameters were uncorrelated,

would require an usually low error in five independent variables. However, in reality we only

require an unusually low error in one parameter and then correlations force all the errors to

be small. Over many noise realisations we would expect the errors in the intrinsic parameters

to wander over the range shown in the posteriors, but this wandering would be correlated

between the different intrinsic parameters.

For the second stage of the search, we ran MULTINEST on the full nine-dimensional

parameter space. We ran a separate analysis, each with 500 ‘live’ points, in the vicinity of

each of the modes identified during the first stage of the search, taking priors on the intrinsic

parametersthatwereλ ∈ [˜λ−5σ,˜λ+5σ], where˜λandσ werethemeanandstandarddeviation

m1and ¯

m2, since the

Page 12

MULTINEST for gravitational wave data analysis 12

tc/yrs

¯

m1/M?

(2.0005 ± 0.0016) × 107

(2.0015 ± 0.0014) × 107

(7.2743 ± 0.0740) × 106

(7.5472 ± 0.0727) × 106

¯

m2/M?

(1.9995 ± 0.0014) × 106

(1.9985 ± 0.0012) × 106

(1.7852 ± 0.0147) × 106

(1.7329 ± 0.0134) × 106

1

1A

2

2A

0.9000 ± 1 × 10−5

0.9000 ± 1 × 10−5

1.0200 ± 2 × 10−4

1.0200 ± 2 × 10−4

θφ

1

1A

2

2A

0.6269 ± 0.0047

2.5131 ± 0.0048

2.1267 ± 0.0060

1.0156 ± 0.0059

4.7312 ± 0.0092

1.5892 ± 0.0087

3.9480 ± 0.0076

0.8076 ± 0.0077

Table 2. Parameter inferences derived from the first, F-statistic, stage of the search. The

quoted uncertainties are the one sigma errors from the posterior recovered by MULTINEST.

We list the parameter inferences for both the true and antipodal (‘A’) sky solutions for each

source.

¯

Mc/M?

1.289 × 103

0.1164

1.198 × 103

0.1336

6.986 × 102

0.7587

7.025 × 102

3.3452

¯ µ/M?

4.719 × 103

0.0763

4.405 × 103

0.0795

8.642 × 103

1.198

8.683 × 103

4.0735

tc/yrs

θφ

1

σFIM

∆λ

σFIM

∆λ

σFIM

∆λ

σFIM

∆λ

1.209 × 10−5

0.0511

1.128 × 10−5

0.9247

3.292 × 10−5

1.1562

3.301 × 10−5

2.7418

6.315 × 10−3

0.0019

9.683 × 10−3

0.1532

6.283 × 10−3

0.9742

6.446 × 10−3

0.3818

1.159 × 10−2

0.2036

8.529 × 10−3

0.6597

7.854 × 10−3

1.0675

7.631 × 10−3

1.0907

1A

2

2A

Table 3. Errors in the intrinsic parameter estimation from the first stage of the search. We

quote errors for both the true and antipodal (‘A’) sky solutions for each source. The first row

for each solution gives the one sigma estimation from the Fisher matrix, σFIM., while the

second row gives ∆λ = |(λT− λR)/σFIM|, where λTdenotes the true parameter value,

and λRdenotes the recovered value of the parameter.

in that parameter as estimated from the posterior recovered in the first stage of the search. In

addition, we took natural priors on the four extrinsic parameters — ι ∈ [0,π], φc ∈ [0,π],

ψ ∈ [0,π] and DL ∈ [2,15]Gpc. If the population of supermassive black hole mergers

was expected to be uniformly distributed in space, then the natural prior on DLwould be

proportional to D2

assembly of structure and will therefore be complicated and very model-dependent. For that

reason, we preferred to use a minimal prior, i.e., a flat prior. This is consistent with the

approach used for the MLDC, in which the source SNRs are drawn from a flat prior [2]. The

prior on φcis [0,π] rather than [0,2π] since there is an angle degeneracy in the extrinsic

subspace corresponding to the shift ψ → ψ + π/2, φc→ φc+ π.

In the second stage of the search, only one mode of the likelihood was found within

L. However, in practice the distribution of events will trace the hierarchical

Page 13

MULTINEST for gravitational wave data analysis 13

the tight priors on the intrinsic parameters provided by the first stage. The second stage of

the search required ∼ 50,000 likelihood evaluations and took ∼ 30 minutes on a single 3.0

GHz Intel Woodcrest processor for each mode. The marginalized posterior distributions for

the true sky locations of the bright and faint sources are shown in Figs. 3 and 4 respectively.

The posteriors for the faint source are very broad in the extrinsic parameters. This is to

be expected, as we do not observe the coalesence for this source and so the error in the

estimate of the phase at coalesence, φc, is very large, and propagates into other parameters

due to correlations. In Table 4 we quote the values of the extrinsic parameters recovered

by the search for the two true modes and the two antipodal modes, and also quote errors as

multiples of the one sigma Fisher Matrix error estimate as before. The antipodal solutions

have different values for ι and ψ as well, which are obtained by the transformation ι → π−ι,

ψ → π − ψ [38], so the true values of the extrinsic parameters at the antipodal sky

location are (ι,ψ,θ,φ) = (2.0296,1.9086,2.5133,1.5708) for the coalescing source,“1”,

and (2.4851,0.5062,1.0210,0.8013) for the non-coalescing source, “2”. We do not include

the intrinsic parameters in this table as their values did not change significantly in the second

stage. The posteriors shown in Figs. 3–4 were computed during the second search stage for all

parameters. We see that the algorithm recovers the extrinsic parameters for both sources and

both sky solutions to the same high precision as the intrinsic parameters. The posteriors are

highly non-Gaussian for the extrinsic parameters of the non-coalescing source in particular, so

the Fisher matrix error should not be trustworthy. However, it appears to be giving predictions

that are largely consistent with the recovered posteriors.

The MULTINEST algorithm also returns the evidence associated with each of the

recovered modes. These log-evidences were 19,948.5 and 19,948.2 for the true and antipodal

modes of the coalescing source and 8,551.3 and 8,555.1 for the true and antipodal modes of

the non-coalescing source. These evidence values do not give very strong reason to favour one

of the two sky-position modes over the other, but the sky position degeneracy is nearly perfect

for the sources we are considering and therefore we would not necessarily expect to be able

to distinguish them. However, the evidences of the 7 other modes identified by MULTINEST

were several hundred lower than the evidences of the true and antipodal modes of the source

to which they corresponded. There would therefore be good reason to favour the true and

antipodal modes over the other modes in the likelihood. If we were using full TDI waveforms

at higher frequency we would expect to be able to distinguish between the true and antipodal

solutions, and this should be reflected in the evidence values. More investigation is needed

to fully understand the utility of evidence for characterisation of modes in the likelihood for

these and other gravitational wave sources.

6. Discussion

We have described the use of a multi-modal nested sampling algorithm, MULTINEST, as

a tool for gravitational wave data analysis, illustrating the algorithm by searching for the

signals from non-spinning supermassive black hole binaries in simulated LISA data. We used

MULTINEST to search a data stream containing two such mergers, and the algorithm was

able to simultaneously and successfully identify both the true sky position of each source, and

the antipodal sky solution, plus several other secondary modes of the likelihood surface. In

a first stage search using the F-statistic, the algorithm is able to recover the intrinsic source

parameters to very high precision, within 2σ of the true answer as measured by the theoretical

noise-induced error computed from the Fisher Matrix. Following up with a search on the full

physical, nine-dimensional, parameter space, the search is also able to recover the extrinsic

parameters to similar precision. In addition, the algorithm naturally recovers the posterior

Page 14

MULTINEST for gravitational wave data analysis 14

Figure 3. Marginalized 1D and 2D posteriors for the primary mode of the bright, coalescing

source (source 1) determined in the second stage analysis using the full likelihood with priors

on the intrinsic parameters derived from the posteriors recovered in the first, F-statistic,

stage. Each plot shows the 2D posterior over the row (vertical axis) and column (horizontal

axis) parameters. At the top of each column is the 1D posterior for that column parameter.

The parameters and the parameter ranges shown in the plots, from top to bottom or left

to right, are 0.63 < θ < 0.64, 4.69 < φ < 4.74, 0.899992 < tc < 0.900008,

1.993 × 107M?< ¯

0.9 < ι < 1.2, 2 < φc < 6 and 1 < ψ < 3 respectively. The true parameter values are

shown by the blue vertical lines and red crosses overlayed on the 1D and 2D marginalized

posterior plots respectively.

m1< 2.007 × 107M?, 1.993 × 106M?< ¯m2< 2.007 × 106M?,

Page 15

MULTINEST for gravitational wave data analysis 15

Figure 4. As Figure 3, but now for the faint, non-coalescing source (source 2). The parameter

ranges shown in the plots, from top to bottom or left to right, are 2.11 < θ < 2.14,

3.93 < φ < 3.97, 1.01988 < tc < 1.02004, 7.1 × 106M? < ¯

1.74 × 106M? < ¯

respectively.

m1 < 7.5 × 106M?,

m2 < 1.81 × 106M?, 0 < ι < 0.8, 0 < φc < 2π and 0 < ψ < π

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- Available from Farhan Feroz · May 26, 2014
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