Addressing out-of-distribution data for flow-based
gravitational wave inference

Maximilian Dax∗
MPI for Intelligent Systems, Tübingenmaximilian.dax@tuebingen.mpg.de

Stephen R. Green∗
MPI for Gravitational Physics, Potsdam
University of Nottingham, Nottingham
stephen.green@aei.mpg.deJonas Wildberger∗
MPI for Intelligent Systems, Tübingen
wildberger.jonas@tuebingen.mpg.de

Jonathan Gair
MPI for Gravitational Physics, Potsdam

Michael Pürrer
MPI for Gravitational Physics, Potsdam
University of Rhode Island, Kingston

Jakob H. Macke
MPI for Intelligent Systems, Tübingen

Alessandra Buonanno
MPI for Gravitational Physics, Potsdam

Bernhard Schölkopf
MPI for Intelligent Systems, Tübingen

Abstract
Simulation-based inference and normalizing flows have recently demonstrated
excellent performance when applied to gravitational-wave parameter estimation.
These methods can provide accurate results within seconds, in cases where classicalmethods based on stochastic samplers may take days or even weeks. However, suchmethods are typically based on deep neural networks and thus unable to reliably
deal with out-of-distribution data, such as may arise when predicted signal andnoise models do not precisely fit observations. We here present two innovations
to deal with this challenge. First, we introduce a probabilistic noise model toaugment the training data, making the inference network substantially more robust
to distribution shifts in experimental noise. Second, we apply importance sampling
to independently verify and correct inference results. This compensates for network
inaccuracies and flags failure cases via low sample efficiencies. We expect thesemethods to be key components for the integration of deep learning techniques into
production pipelines for gravitational-wave analysis.

1

Introduction

Since 2015, the LIGO [1], Virgo [2] and KAGRA [3–5] gravitational wave (GW) observatories havedetected gravitational radiation from 90 astrophysical mergers of black holes or neutron stars [6–8].
Each of these is analyzed using Bayesian inference to compare against predictions from Einstein’s
theory of general relativity and determine the properties of the source. In turn, this has informed ourknowledge of extreme matter and gravity [9, 10], the formation and evolution of binaries [11], andeven the expansion of the universe [12].
∗Equal contribution

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

Observed data in each detector is a time series d, assumed to consist of a GW signal h(θ) and additivenoise n. The signal depends on 15 parameters θ, corresponding to the masses and spins of the binary
components, along with the orientation and location of the binary in space and time. The noise isassumed to be stationary and Gaussian, described by a power spectral density (PSD) Sn , which canvary across detectors and from event to event, and is typically estimated based on data around thetime of the event. Given the pair (d, Sn ), Bayes’ theorem gives the posterior over parameters,p(θ|d, Sn ) =p(d|θ, Sn )p(θ),
p(d|Sn )(1)

and the task of GW parameter estimation is to draw samples from this distribution.
The growing rate of GW detections demands fast sampling techniques to accurately infer parameters
for all observations. Standard inference codes are based on stochastic samplers [13, 14] that require
millions of expensive likelihood evaluations for each event. However, deep-learning techniques haverecently emerged as a promising tool [15–18]. In particular, the D INGO code often matches standard
samplers in terms of accuracy, while being orders of magnitude faster [15, 19].
Artificial neural networks require that training and test data be independent draws from the same
distribution. This is only the case when the measured GW data is consistent with the signal and noisemodels. Since the detector noise PSDs vary with time [20] (e.g., due to detector improvements orvariations of the seismic noise), a network trained with an empiric PSD distribution at the beginningof a LIGO-Virgo-KAGRA (LVK) observing run can only be used for a limited time—once the PSDs
change too much, the measured data become out-of-distribution (OOD). In section 2 we propose aparameterized latent variable model for detector noise PSDs that can be used to augment training
data, enabling D INGO to better adapt to shifting distributions.
Even with accurate modeling of PSD variations one occasionally encounters OOD data due totransient noise artifacts (glitches) or inaccuracies of the signal models [21]. In such cases, theinference network could infer inaccurate results and one would have no means of knowing. In
section 3 we propose to combine neural importance sampling with D INGO. This provides rapid
verification and (in many cases) correction of results, and furthermore flags OOD data for further
investigation. This extended abstract summarizes two recent studies [22, 23]; for further details andextensive empiric results on real data we refer to the original publications.

2

Adapting to noise distribution shifts

D INGO trains a conditional density estimator q(θ|d, Sn ) parameterized with a normalizing flow
[24–26] to estimate p(θ|d, Sn ). A trained network can then be used to draw posterior samples given
any d, Sn consistent with the training distributions. However, the PSD distribution p(Sn ) covering
all future events is unavailable at the time of training. For real-time inference (especially after
experimental upgrades) we therefore propose to use a synthetic PSD distribution q(Sn ) based on past
PSDs and a one-shot observation from an upgraded detector [22].
Methods.—We define the synthetic PSD distribution q(Sn ) as a latent variable modelZ
q(Sn ) = q(Sn |z)qz (z)dz,(2)

where z refers to a set of latent variables that we describe below. We use domain knowledge to defineq(Sn |z) explicitly and integrate causal knowledge about the data generating process. A PSD can be
decomposed
into broad-band noise b (estimated with variance σ 2 ) and a sum of spectral features
Pi si [27], which constitute the main factors of variation between and throughout observing runs.
Thus, our model reads:lXq(Sn |z) = N (b +si , σ 2 ).
(3)
i=1

The latent representation of the broad-band noise is given by y1 , . . . , yk ∈ R+ on a logarithmically
distributed, fixed frequency grid x1 , . . . , xk . b can then be reconstructed by interpolating pairs(x1 , y1 ), . . . , (xk , yk ) using a cubic spline. Each spectral line si is represented by three parameters
f0i , Ai , Qi modeling the center, amplitude and width of a truncated Cauchy distribution, respectively [27]. We segment the frequency space into l equally wide sub-intervals and model a single
2

0.05

O3a

O3b

4

q

0.

8

0.03

0.

0.02

6

JSD [nat]

0.04

LALInference
Dingo-IS
Dingo

×10−3

250

300

350

Oracle noise

Synthetic noise

Naive noise

Mc [M ]

q

α

6

200

days

3

150

8

100

0.

50

4

0

0.

0.00

24

α
3

0.0

18

4.5

0.01

Figure 1: Left: Time development of D INGO performance since the beginning of O3 (Day 0) for
models trained with different PSD datasets. With our Synthetic PSDs, we achieve comparable
performance to the model trained with Oracle PSDs, despite only using a single O3 PSD. Right:
Chirp mass (Mc ), mass ratio (q) and sky position (α) parameters for GW151012. Even when initial
D INGO results (black) deviate from LALI NFERENCE -MCMC posteriors (blue), importance samplingleads to close agreement (orange).

spectral line in each of them.2 In summary, the latent representation of a PSD is given by parametersz = (y1 , . . . , yk , f01 , A1 , Q1 , . . . f0l , Al , Ql ) ∈ Rk+3l .

We first project all PSDs from past observing runs onto their latent representation z using maximumlikelihood estimates. We then use these latent samples to fit Gaussian kernel density estimates(KDEs) over independent noise sources, modelling a disentangled latent distribution qz (z). The
disentanglement enables meaningful interventions on the latent space, which we leverage to adjust the
distribution for future data. Indeed, we first use a single PSD at the start of an observing run to rescale
the mean of qz (y1 , . . . , yk ) to account for increased detector sensitivity. And then we broaden thelatent distribution qz (z) by increasing the KDE bandwidth to account for uncertainty in the estimatedmodel and fluctuations in the latent features that may occur over the course of an observing run. With
this approach, we can prepare D INGO networks for an entire observing run without having access tofuture PSDs.
Results.—We evaluate our approach in a real-world setting by reanalyzing the third LVK observing
run (O3). As a baseline, we train a D INGO model only with PSD data from the first four days of O3;
this Naive model would be available shortly after the start of O3. We also train a D INGO model usinga Synthetic PSD dataset from q(Sn ) based on past PSDs and scaled by the first PSD of O3. Finally,
we train an Oracle model based on PSDs from all of O3 (≈ 1 year), which upper bounds the potential
performance of our noise model.
We first analyze a simulated GW signal injected into Gaussian noise drawn from a series of O3 PSDs.
As a performance metric, we compare the inferred D INGO results to reference posteriors generated
with D INGO -IS (see section 3) in terms of the mean Jensen-Shannon divergence (JSD) of the 1D
marginals. Fig. 1 shows the time development of the D INGO performance. Indeed, the accuracy of the
Naive baseline degrades with time. The Oracle model shows consistent performance throughout O3,
however, in reality such a model could only be trained at the end of O3. D INGO models trained with
the Synthetic PSD dataset almost match the Oracle performance, enabling accurate online analysis.
Finally, we analyze 37 real GW events from O3. Averaging across all events (except for an outlier
event GW190517_055101) and parameters, our Synthetic PSD dataset achieves a mean JSD of
1.4 · 10−3 nat, which is only slightly worse than with the Oracle PSDs (1.2 · 10−3 nat) and far
2This design choice restricts the number of spectral lines per segment to zero (via a vanishing amplitude
parameter) or one. More than one spectral line cannot be modeled, but we found this to be irrelevant in practice
if l is sufficiently large.

3

Sample Efficiency
D INGOD INGO-ISB ILBYD INGOD INGO-ISB ILBY

28.8%
0.14%
12.5%
0.16%

Mean JSD
2.2
0.5
1.8
9.0
0.7
1.1

Max JSD
7.2 (α)1.4 (dL )4.0 (dL )53.4 (Mc )2.2 (α)4.1 (α)log p(d)−15831.87 ± 0.01
−15831.78 ± 0.10−16412.88 ± 0.01
−16412.73 ± 0.09

Table 1: D INGO performance for GW150914 (upper block) and GW151012 (lower). The JSD quantifies the deviation to LALI NFERENCE -MCMC, all values in 10−3 nat. The mean is taken across allparameters. Results with a maximum JSD ≤ 2×10−3 nat are considered indistinguishable [32]. Here,maxima occur for right ascension α, luminosity distance dL , and chirp mass Mc . For comparison, we
also report results from B ILBY- DYNESTY.

outperforms the Naive baseline (2.7 · 10−3 nat). This shows how the generative PSD model can beused to enhance the generalization capability of D INGO.

3

Neural importance sampling

Even accounting for shifts in detector PSDs, observed GW data can still be OOD if the noise is
non-Gaussian or the real signal is inconsistent with signal models. We here propose to combine
amortized neural posterior estimation (NPE) [28]—the method underlying D INGO—with importancesampling. This extension (“D INGO -IS” [23]) provides interpretable diagnostics to flag failure cases
and asymptotically recovers the true posterior. Establishing this network-independent verificationand correction mechanism makes D INGO substantially more reliable for dealing with real data.
Methods.—Given a set of n samples θi ∼ q(θ|d, Sn ) ≡ q(θ|d), we assign each an importance weight
wi = p(d|θi )p(θi )/q(θi |d). These can be computed since the likelihood and the D INGO proposal areboth tractable. If D INGO samples matched theperfectly, then wi = constant. MoreP true posterior
P
generally, the effective sample size is neff = ( i wi )2 / i wi2 and we can quantify the quality of the
proposal with thePsample efficiency ϵ = neff /n. Further, the Bayesian evidence p(d) can be estimatedas p(d) = 1/n i wi . The variance of p(d) scales with 1/n allowing for very precise estimates.
Importance sampling asymptotically recovers the exact posterior if the proposal distribution is masscovering, i.e., supp(p(d|θ)p(θ)) ⊆ supp(q(θ|d)). This is ensured by D INGO (and NPE in general)
through minimizing the forward KL-divergence DKL (p(θ|d)||q(θ|d)) during training, which diverges
unless q(θ|d) indeed covers the entire posterior probability space. This property is not guaranteed byclassical stochastic samplers or by other machine learning methods optimizing different objectives(e.g., variational inference [24, 29]), so NPE is particularly well suited for importance sampling.
Although importance sampling requires likelihood evaluations at inference time, a high sample
efficiency (due to a high-quality D INGO proposal) and parallelizability make it much faster thanother likelihood-based methods. The combination of NPE and importance sampling is an amortized
extension of neural importance sampling [30]. We anticipate this to be a useful approach beyond GW
science to verify inference results of deep learning methods.
Results.—We first validate D INGO -IS on two real GW events (GW150915 and GW151012) bycomparing against LALI NFERENCE -MCMC [13], an established GW inference code. In particular,
for GW151012, we find slight disagreement between D INGO and LALI NFERENCE (Fig. 1 andTab. 1). Compared to [15], we use the more complicated waveform model IMRPhenomXPHM [31]and a larger prior, so small D INGO inaccuracies are not surprising. The importance sampled D INGO
result, however, is in excellent agreement with LALI NFERENCE. For the log evidences, we findgeneral agreement between the D INGO and B ILBY- DYNESTY [14, 32, 33] estimates.
We also conduct a large scale study using two waveform models (IMRPhenomXPHM [31] andSEOBNRv4PHM [34]) analyzing 42 events from O3. Running stochastic samplers with SEOBNRv4PHM requires several months of computation per event, so a study of this scale is only feasible
due to the superior speed of D INGO (≈ 20 seconds per event) and D INGO -IS (≈ 10 h per event
for SEOBNRv4PHM; < 1 h for IMRPhenomXPHM). Across all events, we find a median sample
4efficiency of ϵ = 10.9% for IMRPhenomXPHM and ϵ = 4.4% for SEOBNRv4PHM. This is twoorders of magnitudes larger than the sample efficiencies achieved by stochastic samplers, which areon the order of 0.1% (see Tab. 1). For most events, we find good agreement between D INGO andD INGO -IS, indicating high-quality inference results. Moreover, many events where D INGO performs
poorly are known OOD events: nine events are known to suffer data quality issues (specifically, glitch
artifacts), and for most of these, D INGO -IS has very low ϵ. Indeed, deep learning methods are known
to perform poorly when the data does not match the training distribution. A low sample efficiency
thus arises because the deep-learning extrapolation away from the training distribution differs fromthe posterior as defined by the specified prior times likelihood in this region. We rely on this discrepancy to identify potential OOD data. Likewise, adversarial attacks [35, 36] are also flagged withextremely low sample efficiency (ϵ ≈ 0.01%) [23]. This showcases the use of importance samplingin conjunction with deep learning methods to flag potential failure cases such as OOD data.
Experimental details.—Here and in section 2 we use the setup from [15]. D INGO models are trainedfor ≈ 1 week on an A100 GPU and 32 CPUs. For D INGO -IS, we use 64 CPUs for the likelihood
evaluations. We use PyTorch [37], nflows [38] and the Adam optimizer [39], which are all freely
available under MIT or BSD license.

4

Conclusion

In this work, we proposed two approaches to deal with OOD data for flow-based GW parameter
estimation, making D INGO into a more reliable and versatile parameter estimation tool. First, we
introduced a probabilistic framework to model variations of the detector noise PSDs. We empiricallydemonstrated that this greatly enhances the generalization capabilities of D INGO to drifting detector
noise distributions—enabling real-time analysis with a network trained at the beginning of anobserving run. Second, we augmented D INGO with importance sampling (D INGO -IS). This provides
useful diagnostics for detecting OOD data, and at the same time corrects potentially inaccurateinference results. D INGO -IS has a substantially larger sample efficiency than stochastic samplers
and is fully parallelizable, resulting in great speed advantages. It also provides Bayesian evidence
estimates with ten times greater precision, enabling detailed model comparison. Going forward, we
expect these techniques to become key components for the integration of D INGO into production GW
parameter estimation pipelines.

Broader impact statement
Our methods are primarily targeted at scientific applications, and we do not foresee direct applications
which are ethically problematic. In the context of GW analysis, we hope this study helps to establish
efficient deep learning methods in production pipelines. This could reduce the required amount ofcompute compared to standard inference methods, in particular when the rate of detections increases
with more sensitive detectors in the future.

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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] See Section 2 and 3(b) Did you describe the limitations of your work? [Yes](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...
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(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] Our code iscurrently reviewed internally by the LVK collaboration. We will make it publicallyavailable, once the review process is concluded.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [Yes] See end of section 3, which also refers to [15].
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