Fast likelihood evaluation using meshfree approximations for reconstructing compact binary sources Lalit Pathak1Amit Reza2 3and Anand S. Sengupta1

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Fast likelihood evaluation using meshfree approximations

for reconstructing compact binary sources

Lalit Pathak,

1, ∗

Amit Reza,

2, 3, †

and Anand S. Sengupta

1, ‡

Indian Institute of Technology Gandhinagar, Gujarat 382355, India.

Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands.

Institute for Gravitational and Subatomic Physics (GRASP),

Utrecht University

, Princetonplein 1, 3584 CC Utrecht, The Netherlands.

Several rapid parameter estimation methods have recently been advanced to deal with the

computational challenges of the problem of Bayesian inference of the properties of compact binary

sources detected in the upcoming science runs of the terrestrial network of gravitational wave

detectors. Some of these methods are well-optimized to reconstruct gravitational wave signals in

nearly real-time necessary for multi-messenger astronomy. In this context, this work presents a new,

computationally eﬃcient algorithm for fast evaluation of the likelihood function using a combination

of numerical linear algebra and mesh-free interpolation methods. The proposed method can rapidly

evaluate the likelihood function at any arbitrary point of the sample space at a negligible loss of

accuracy and is an alternative to the grid-based parameter estimation schemes. We obtain posterior

samples over model parameters for a canonical binary neutron star system by interfacing our fast

likelihood evaluation method with the nested sampling algorithm. The marginalized posterior

distributions obtained from these samples are statistically identical to those obtained by brute force

calculations. We ﬁnd that such Bayesian posteriors can be determined within a few minutes of

detecting such transient compact binary sources, thereby improving the chances of their prompt

follow-up observations with telescopes at diﬀerent wavelengths. It may be possible to apply the

blueprint of the meshfree technique presented in this study to Bayesian inference problems in other

domains.

I. INTRODUCTION

The detection of gravitational waves (GW) from the

GW170817 [

] binary neutron star (BNS) system, followed

by the prompt multi-wavelength (gamma-rays to radio)

observation of its electromagnetic (EM) counterpart, has

led to several fundamental discoveries; and is hailed as a

signiﬁcant breakthrough in astronomy. These discoveries

include the validation of long-held hypotheses that BNS

mergers are ideal sites for r-process nucleosynthesis and

produce short gamma-ray bursts, the ﬁrst GW-based

constraints on the equation of state of nuclear matter

in such stars, and the measurement of Hubble constant

independent of the cosmic distance ladder.

The inevitable improvement of the detectors’ sensitiv-

ity in future observation runs is likely to have a two-fold

impact on the prospects of multi-messenger observations:

ﬁrstly, the increased bandwidth of the detectors (espe-

cially improved sensitivity at low frequencies) will result

in a tremendous increase in the computational cost of

∗lalit.pathak@iitgn.ac.in

†areza@nikhef.nl

‡asengupta@iitgn.ac.in

Bayesian inference of source parameters, including sky lo-

calization essential for prompt observation of EM counter-

parts. Although the

BAYESTAR

[

] algorithm could be used

to produce rapid sky maps, it has been recently shown [

]

that coherent parameter estimation (PE) can localize the

sources better by an average reduction of 14

deg2

in the

uncertainty, underlining the importance of developing

fast PE algorithms. Second , the reach of the terrestrial

network of GW detectors will extend out to several Gpc

to the eﬀect that one would have far too many detections

of BNS/NSBH signals to contend with whilst generating

prompt sky-location maps [

]; so much so that one may

have to prioritize the GW sources for EM follow-up based

on prospects of new science from a rapid estimation of

their mass and spin components as shown by Margalit

&Metzger [

], thereby helping EM observatories to use

resources optimally. Several fast PE algorithms have been

developed recently, such as the coherent multi-detector ex-

tension of the relative binning/heterodyne method by Fin-

staad and Brown (2020) [

], which produces the posterior

within twenty minutes for BNS systems with 32 CPU cores.

Well-trained machine learning PE methods [

–

] can sig-

niﬁcantly reduce the runtimes and produce the posteriors

in nearly real-time. In the past, algorithms for accelerated

parameter estimation have mainly focussed on speeding

arXiv:2210.02706v2 [gr-qc] 1 Dec 2023

up the overlap integral. These include reduced-order mod-

els (ROMs) [

–

], machine-learning aided ROMs [

Gaussian process regression based interpolation [

] and

relative binning [

] algorithms. Our approach takes

inspiration from the grid-based likelihood interpolation

method [

] based on orthonormal Chebyshev polynomi-

als. The grid-based techniques have a drawback in that

the number of interpolation nodes grows exponentially

with the dimensionality of the parameter space.

In this work, we propose a new and alternative

approach to grid-based likelihood interpolation method

[

], a computationally eﬃcient method for evaluating the

likelihood function (a key ingredient in Bayesian inference)

using meshfree interpolation methods with dimension

reduction techniques. We directly interpolate the

likelihood function over the parameter space, bypassing

the generation of templates and brute-force computation

of the overlap integral altogether. Our scheme can

quickly approximate the log-likelihood function with

high accuracy and produce statistically indistinguishable

posteriors over source parameters. Further, both the

GstLAL

search framework [

] and the meshfree method

use the idea of dimension reduction using SVD [

], it

may be prudent to incorporate this method with the

low-latency

GstLAL

search pipeline for rapid, automated

follow-ups of the detected events.

II. BAYESIAN INFERENCE

Given data

d=h(⃗

Λtrue) + n

recorded at a detector

containing an astrophysical GW signal

(

⃗

Λtrue

)embedded

in additive Gaussian noise

, one is interested in solving

the inverse problem to estimate the source parameters.

Bayesian inference is a stochastic inversion method where

the posterior probability density

(

⃗

Λ|d

)over the source

parameters is related to the likelihood function

L(d|⃗

Λ)

of observing the data through the Bayes’ theorem:

p(⃗

Λ|d) = L(d|⃗

Λ) p(⃗

Λ)

p(d),(1)

where

(

⃗

)is the prior distribution over the model

parameters

⃗

Λ≡ {⃗

λext,⃗

λ}

. In our notation,

⃗

denotes

the intrinsic parameters such as component masses and

spins. The set of extrinsic parameters is denoted by

⃗

λext

We are particularly interested in estimating the extrinsic

parameter

denoting the ﬁducial time of coalescence of

the two masses.

will be mentioned explicitly wherever

required, as it is treated in a special way in our analysis.

The forward generative frequency-domain restricted

waveform model for non-precessing compact binaries can

be expressed as

h(⃗

Λ) = Ah+(fk;⃗

λ)

, where the complex

amplitude

depends only on the extrinsic parameters

and

(

;

⃗

)is the ‘+’ polarization of the signal that

depends only on the intrinsic parameters [

]. Here

{fk}Ns/2

k=0

deﬁnes positive Fourier frequencies, and

Ns1

is the number of sample points. A GW signal, observed

by an interferometric detector, can be considered as a

linear combination of the two polarizations weighted by

the antenna pattern function. The

(

;

⃗

)polarization

is related to the

h×

(

;

⃗

)polarization for non-precessing

GW signal as:

h+(fk;⃗

λ)∝ih×(fk;⃗

λ)

[

]. This allows

us to write the detector response in terms of any one

of the polarizations alone (we have chosen the

(

;

⃗

)

polarization). Using this model, the posterior

p(⃗

Λ|d)

can

be directly evaluated at every point in

⃗

using Eq. (1).

However, in view of the high-dimensionality of

⃗

, it is

more eﬃcient to sample the posterior using stochastic

sampling algorithms such as Nested-Sampling [

], or

Markov Chain Monte Carlo (MCMC) [

]. From Eq. (1),

it is evident that for a quick estimation of the posterior

distribution, it is imperative to rapidly evaluate the

likelihood function.

We work with the phase-marginalized log-likelihood

function [22]:

ln L(⃗

Λ, tc) = ln I0h|A| z(⃗

λ, tc)i−1

2∥h(⃗

Λ)∥2

2(2)

where

(

)is the 0-th order modiﬁed Bessel function

of the ﬁrst kind, and

(

⃗

λ, tc

)is the frequency-domain

overlap-integral:

z(⃗

λ, tc) = 4 ∆f



Ns/2

k=0

d∗(fk)h+(fk,⃗

λ)

Sh(fk)e−2πifktc



,(3)

inversely by

(

), the detector’s one-sided noise power

spectral density (PSD). The data and template vectors

are sampled at discrete frequencies {fk}Ns/2

k=0 .

The complexity of evaluating the overlap integral scales

directly with the number of data samples, which in

turn, scales with the seismic cut-oﬀ frequency (approx-

imately) as

Ns∼f−8/3

low

. As we progress from the O4

observational run (

flow = 20 Hz

) to O5 at design sensi-

tivity (

flow = 10 Hz

), evaluating

p(⃗

Λ|d)

is likely to take

at least

3longer. In addition, additional costs will

be incurred in constructing longer templates at the pro-

posal points. Therefore, the likelihood calculation can

be expensive. However, our method is immune to this

issue as our scheme directly approximates the likelihoods

1Ns=signal duration ×sampling frequency

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FastlikelihoodevaluationusingmeshfreeapproximationsforreconstructingcompactbinarysourcesLalitPathak,1,∗AmitReza,2,3,†andAnandS.Sengupta1,‡1IndianInstituteofTechnologyGandhinagar,Gujarat382355,India.2Nikhef,SciencePark105,1098XGAmsterdam,TheNetherlands.3InstituteforGravitationalandSubatomicPhysics(GR...

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Fast likelihood evaluation using meshfree approximations for reconstructing compact binary sources Lalit Pathak1Amit Reza2 3and Anand S. Sengupta1

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