Amortized Bayesian inference of gravitational waves
with normalizing flows

Maximilian Dax
MPI for Intelligent Systems,
72076 Tübingen, Germany
maximilian.dax@tuebingen.mpg.de

Stephen R. GreenMPI for Gravitational Physics,
14476 Potsdam, Germany
stephen.green@aei.mpg.de

Jonathan Gair
MPI for Gravitational Physics,
14476 Potsdam, Germany

Jakob H. Macke
MPI for Intelligent Systems,
72076 Tübingen, Germany

Alessandra Buonanno
MPI for Gravitational Physics,
14476 Potsdam, Germany

Bernhard Schölkopf
MPI for Intelligent Systems,
72076 Tübingen, Germany

Abstract
Gravitational waves (GWs) detected by the LIGO and Virgo observatories encode
descriptions of their astrophysical progenitors. To characterize these systems,
physical GW signal models are inverted using Bayesian inference coupled withstochastic samplers—a task that can take days for a typical binary black hole.
Several recent efforts have attempted to speed this up by using normalizing flows toestimate the posterior distribution conditioned on the observed data. In this study,
we further develop these techniques to achieve results nearly indistinguishable fromstandard samplers when evaluated on real GW data, with inference times of one
minute per event. This is enabled by (i) incorporating detector nonstationarity fromevent to event by conditioning on a summary of the noise characteristics, (ii) using
an embedding network adapted to GW signals to compress data, and (iii) adopting
a new inference algorithm that makes use of underlying physical equivariances.

1

Introduction

Since 2015, the LIGO and Virgo gravitational-wave observatories [1, 2] have together detected signals
from over 50 mergers of black holes and/or neutron stars [3–5]. Given a detection, the data arecompared against theoretical predictions using Bayesian inference to characterize the system. In thisstudy we focus on inference of binary black hole (BBH) mergers, as these are most common andhave shorter observable signals than mergers involving neutron stars.
Signal models take the form of time-series waveform predictions h(θ) as a function of systemparameters θ (including the black-hole masses and spins, the orientation of the binary, and its
localization in space and time). These are made on the basis of Einstein’s theory of general relativity.
To this is added detector noise n, assumed to be stationary and Gaussian to a good approximation[although, importantly, the noise power spectral density (PSD) Sn can vary from event to event]. This
gives the data generative process (or likelihood) p(d|θ, Sn ), where d = h(θ) + n. Combined with aprior, Bayes’ theorem gives the posterior distribution over parameters,p(θ|d, Sn ) =p(d|θ, Sn )p(θ),
p(d|Sn )Fourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

(1)

where the normalization factor p(d|Sn ) is the Bayesian evidence.
Standard codes for gravitational-wave inference [6, 7] use stochastic methods such as Markov chain
Monte Carlo (MCMC) to draw samples from the posterior. Typically, ∼ 104 posterior samples
are desired, which requires millions of likelihood evaluations (waveform simulations). Even using
highly optimized models, this can take days per event. With an ever-increasing event rate due toimprovements in detector sensitivity, the computational cost of analyzing all events is becoming verysignificant.
Many studies have attempted to address the challenges of GW inference using machine learning [8–15]. These approaches all give fast inference, but the challenge remains to produce accurate andcomplete results competitive with conventional algorithms. The present extended abstract (a workshop
version of [16]) describes an approach that finally moves beyond proof-of-concept by producing
results nearly indistinguishable from conventional algorithms. We build upon a previous study [12]by some of us using neural posterior estimation (NPE) [17] with conditional normalizing flows toestimate the GW posterior. However, we now achieve full amortization of training costs acrossobservations by accounting for detector noise nonstationarity from event to event. We also introduce
an embedding network to treat the high-dimensional GW data sets, and we develop a new algorithm toincorporate model equivariances into our framework. We demonstrate our approach (called “D INGO”)
by analyzing eight GW events from the first Gravitational-Wave Transient Catalog (GWTC-1) [3],achieving excellent quantitative agreement with standard codes. Inference on each event with afully-trained network takes approximately one minute.

2

MethodD INGO trains a flexible conditional density estimator q(θ|d, Sn ) to approximate p(θ|d, Sn ). We use anormalizing flow [18–20], to define q(θ|d, Sn ) via a mapping fd,Sn : u 7→ θ from a standard-normal“base” distribution N (0, 1)D (u),−1q(θ|d, Sn ) = N (0, 1)D fd,S(2)
(θ)
det Jf −1 ,nd,Snwhere D = 15 is the dimension of the parameter space. This is simply the change-of-variables rule
for probability distributions. fd,Sn is chosen to be invertible with simple Jacobian determinant, soq(θ|d, Sn ) can be rapidly evaluated and sampled from. Our basic approach is based on that of [12],and we now highlight the main improvements; for further details see [16].
2.1Noise conditioning

Noise in GW detectors is to a good approximation stationary and Gaussian. However, it does vary
to some degree from event to event, and this must be taken into account for inference at the desired
accuracy. The noise PSD is typically estimated from signal-free detector data in the vicinity of anevent, and it enters into conventional analyses via the likelihood. For NPE, we use the PSD to whiten
the strain data before passing it to the flow. However, as shown in [16], this is not sufficient to fully
account for PSD drifts, since information is lost if the PSD is discarded after whitening.1 We resolve
this problem by providing the PSD as additional context information to the flow.
Past studies applying similar approaches to GWs [8–15] used a fixed PSD for training. In this study,
we augment our training data with a collection of PSDs estimated during LIGO/Virgo observing runs(∼ 103 PSDs per detector). These are drawn randomly during training, and are used to generate noise
realizations that are added to simulated signals. At inference time, the estimated PSD at the time ofthe event is provided as context along with the signal data.
2.2

Embedding network

We represent data in the frequency domain since this is the natural representation for stationary
Gaussian noise PSDs. For 8 s waveforms (suitable for BBHs), and frequencies between 20 and
1
For instance, when working with whitened data, scaling the PSDs in all detectors with an overall factor α2
has an effect equivalent to scaling the luminosity distance dL by α. If no information about the PSD is provided
to the inference network this degeneracy impedes correct inference of dL .

21024 Hz, we have 24,096 (real) input dimensions per detector. This must be heavily compressedusing an embedding network before conditioning the flow. We first apply a linear layer to project thedata to 400 components per detector. To provide an inductive bias to extract signal information, we
initialize this with the principal components [obtained via a singular value decomposition (SVD)] ofclean waveforms from our training set. Implementing the SVD compression with a learnable layer(instead of a fixed projection as done in previous works) allows the inference network to learn a more
useful representation while retaining the inductive bias provided by the SVD initialization. Following
this, a fully-connected residual network [21] compresses to 128 features.
The flow itself is a composition of 30 rational-quadratic spline coupling flows [22], each of which ismade up of 5 two-layer residual blocks, and is conditioned on the output of the embedding network.
The embedding network and flow are trained alongside each other.
2.3

Group equivariant neural posterior estimation

One of the main performance impediments encountered in our experiments had to do with estimating
the time of binary coalescence tI measured in each detector I. The relative time of arrival of thesignal in each detector is related through triangulation to the sky position of the source, and we alsoinfer the overall time of coalescence at geocenter, so the prior includes training data with varyingtI . In frequency domain, time translations correspond to local phase shifts, which, although wellunderstood, can be challenging for neural networks to learn based on simulations alone. Indeed, this
occupied much of the network capacity in [12].
If we had precise knowledge of tI then we could manually time shift the data to a fixed coalescence
time to simplify the task of the network. However, tI = tI (θ) is a function of the parameters θ,which are a priori unknown at inference time. Our new approach—called group equivariant neural
posterior estimation (GNPE) [23]—resolves this problem. GNPE is a general method that enables toself-consistently apply θ-dependent transformations to observational data d despite unknown θ. As
discussed in detail in [23], this can be used to integrate exact equivariances of a forward model byconstruction, and also allows for approximate equivariances.
In the context of GW parameter inference GNPE works as follows. For each θ-dependent variable tI
we define a proxyt̂I = tI (θ) + ,

 ∼ κ()(3)

as a perturbed version of tI . We choose a narrow, uniform kernel κ = U [−1 ms, 1 ms] for the
perturbation , therefore the proxy t̂I is a good approximation of tI . With that definition, we aim to
sample from the extended posterior2θ, t̂ ∼ p θ, t̂|d ,(4)
where t̂ is a short hand notation collecting the proxies t̂I for all detectors I. Samples from (4) can beeasily turned into samples θ ∼ p(θ|d) from the desired posterior by simply marginalizing over t̂ (i.e.,
dropping the corresponding axes). As becomes apparent below, it is easier to learn a representationof the extended posterior p(θ, t̂|d) than to directly learn p(θ|d), since the inference network for theextended posterior can be trained on GW data with (almost) no time shifts.
With GNPE, we infer the extended posterior (4) with Gibbs sampling [24, 25]. Specifically, we
iteratively sample θ and t̂ conditioned on the respective other parameter as well as on d,t̂ ∼ p t̂|d, θ⇐⇒t̂ = t(θ) + ,  ∼ κ(),(5)θ ∼ p θ|d, t̂⇐⇒θ ∼ q θ|d−t̂ , t̂ .
(6)Here the left-hand side describes the sampling operation and the right-hand side how it is performed inpractice. While sampling t̂ for given θ in (5) is trivial with its definition (3), we employ a normalizing
flow q in (6) to estimate p θ|d, t̂ and sample θ. Importantly, due to the conditioning on t̂ (and theinvertibility of time shifts), we can time shift the strain data by −t̂ before providing it to the flow,which we denote with d−t̂ . The normalizing flow is thus trained with simulations with almost no
time shift, since tI − t̂I is restricted to the range [−1 ms, 1 ms] of the kernel κ. We found that thisre-alignment of the data d is crucial for the accuracy of the flow.
2

For readability, we leave the conditioning of the posterior on the PSD Sn implicit in this subsection.

3

χeff

q

− −
0 0 0 0 0 0 0 0 025 50 .75 .30 .15 .00 .15.45 .60 .75 .90

4h

0.

8h

M [M ]

q

χeff

75

50

0.

25

0.

0.

3
0. 4
45
0.
60
0.
75
0.
− 90
0.
− 30
0.
1
0. 5
00
0.
15

32

30

28

0.

χp

12h

χp

(a)

(b)

Figure 1: Comparison of GW150914 posterior distributions produced by LALI NFERENCE MCMC
(blue) and D INGO (orange). Contours represent 50% and 90% credible regions. Panel (a) shows the
chirp mass M, mass ratio q, and two effective spin parameters χeff , χp ; (b) shows the sky position.
Iterative application of (5) and (6) results in a Markov chain that asymptotically converges to theextended posterior (4). In practice, we speed up convergence by constructing N Markov chains to
obtain N samples in parallel. We initialize each of these by sampling from a flow q(tI |d) trainedwith standard NPE to infer initial tI estimates. From each of the N Markov chains we extract only
the final sample. As shown in [23], this significantly improves the convergence behaviour, such that
only ∼ 30 GNPE iterations are required.
Training
The GNPE inference network is trained by minimizing the loss
L = Ep(θ) Ep(Sn ) Ep(d|θ,Sn ) Eκ() − log q θ|d−(t(θ)+) , Sn , t(θ) +  .

(7)

(i)
Estimating (7) requires sampling θ(i) ∼ p(θ) and Sn ∼ p(Sn ), and then simulating data d(i) ∼(i)p(d|θ(i) , Sn ). We choose a prior suitable for BBH systems, with uniform component masses

m1 , m2 ∈ [10, 80] M , luminosity distance dL ∈ [100, 6000] Mpc, and standard uninformative
priors for sky position, orientation, and spins. We train separate networks for detector noise levelsin the first (O1) and second (O2) observing runs of LIGO and Virgo, with PSD samples estimatedempirically from noise data [26]. As in [12], training data are generated from a fixed set of spinprecessing frequency-domain waveforms, described by the IMRPhenomPv2 [27–29] model, but withextrinsic parameters and noise realizations drawn randomly during training. We train for 450 epochs
using the Adam optimizer [30], and reserve 2% of the training data for validation. With training setsof 5 × 106 waveforms, there is no indication of overfitting. Training takes between 16 and 18 days
on a single NVIDIA Tesla V100. We use PyTorch [31] and nflows [32] for the implementationof our neural networks. The plots are generated with ChainConsumer [33] and ligo.skymap [34].
All software used in this project is freely available under MIT or BSD license.

3

Results

We evaluate D INGO on all GWTC-1 events with component masses greater than 10 M (ourprior bound). For all events, we use the data from the two LIGO detectors for the analysis; for
GW170814 (the first 3-detector event) we additionally use data from the Virgo detector. We compare4

Table 1: Deviation between D INGO and LALI NFERENCE posteriors, quantified by the JensenShannon divergence (JSD) between 1D marginals. We report the mean JSD across all parameters, andthe maximal JSD, including the corresponding parameter. All JSDs are reported in units of 10−3 nat.
event

mean JSD

max JSD

event

mean JSD

max JSD

GW150914
GW151012
GW170104
GW170729

0.6
0.7
1.0
1.3

1.4 (δ)2.7 (m1 )
6.4 (m1 )
6.3 (dL )GW170809
GW170814
GW170818
GW170823

0.9
1.0
1.3
0.4

5.5 (δ)2.5 (α)3.8 (α)0.9 (dL )against MCMC posteriors produced using the standard LIGO/Virgo parameter estimation code
LALI NFERENCE [6]. For D INGO, generation of 50,000 sample points with 30 GNPE iterations takes
roughly 1 minute. A qualitative comparison for the first GW detection is displayed in Fig. 1, showing
excellent overlap.
For quantitative comparisons, we compute the Jenson-Shannon divergence (JSD) [35] between
1D marginalized posteriors, a divergence which lies between 0 and ln(2) ≈ 0.69 nat. Across all
events and parameters,3 we find a mean JSD of 0.0009 nat, which is only slightly higher than thevariation (0.0007 nat) found between LALI NFERENCE runs with identical settings but different
random seeds [37]. Moreover, for GW samplers, a maximum JSD across parameters of 0.002 nat isregarded as indistinguishable [37]; our results approach this, with two events below for all parameters,
and the others with one to three parameters above; see Tab. 1. For comparison, variations in the PSD
and the choice of waveform model [37] both impact the JSD at a much higher level of ≈ 0.02 nat.

4

Conclusions

In this work, we achieved unprecedented accuracy for rapid GW parameter inference using normalizing flows. We analyzed eight GWTC-1 events, and showed excellent agreement with standard codes,
with inference times reduced by over three orders of magnitude. This improvement in performancecompared to past studies was achieved by conditioning on the noise PSD, introducing a powerful
embedding network, and using the novel GNPE algorithm to simplify the learning task by aligning
the signal waveforms in each detector. The latter is a general approach to treating equivariances,which we hope will be useful in other inference applications as well.
The present study is limited to analyzing BBH systems, with relatively simple waveform models, andwith stationary Gaussian noise. To extend to longer signals for binary neutron stars and more complex
signals incorporating more physics [38] will require somewhat larger networks and improved data
representation or compression. D INGO should scale better than conventional algorithms to expensive
waveforms since generation of training data can be fully paralellelized. Moreover, as a likelihood-freemethod, it does not impose any fundamental restrictions on the noise model and should ultimately
reduce systematic errors of current analyses (although in this study we used stationary Gaussian noise
to compare against conventional samplers). We plan to address these limitations in future studies.
Deep learning is now able to analyze the vast majority of LIGO/Virgo events at comparable accuracy
to standard algorithms. Through future extensions we expect that D INGO could become one of theleading approaches to GW inference.

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes](b) Did you describe the limitations of your work? [Yes] See section 4.
(c) Did you discuss any potential negative societal impacts of your work? [N/A](d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A](b) Did you include complete proofs of all theoretical results? [N/A]
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
[No] We plan to release a production-level python package for D INGO in the very
near future, including scripts to reproduce our results. The current research code has
substantial dependencies on local infrastructure.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)?
[Yes] The most important details are given in section 2, for full information we refer tothe corresponding paper [16].
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
[No] The computational cost of training prohibits multiple identical runs.
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)? [Yes] See end of section 2
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] End of section 2.
(b) Did you mention the license of the assets? [Yes] See end of section 2.
(c) Did you include any new assets either in the supplemental material or as a URL?
[No] All code and models will be included in the D INGO package, see question 3(a).
(d) Did you discuss whether and how consent was obtained from people whose data you’reusing/curating? [N/A](e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
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