1 Introduction
The number of detected gravitational wave events is rapidly increasing abbott2019gwtc ; nitz20191 ; zackay2019highly ; nitz20202 ; venumadhav2020new ; abbott20211gwtc ; abbott20212gwtc . Consequently, datasets of inferred properties such as mass, spin, and redshift have become feasible, the most recent being produced from GWTC3 abbott20212gwtc . At this point, there is sufficient data for populationlevel analyses kovetz2017black ; fishbach2018does ; wysocki2019reconstructing ; roulet2019constraints ; smith2020inferring ; galaudage2020gravitational ; kapadia2020self ; kimball2020black ; roulet2020binary ; tiwari2021vamana ; abbott2021population ; abbott2021population3 . Many previous methods assume a simplistic phenomenological shape of the distribution such as a power law. However, as indicated in tiwari2021vamana , it is crucial that the proposed model class can accurately represent the data; otherwise, the reconstruction may miss essential populationlevel properties and correlations. An imposed phenomenological shape can suffer from humaninduced bias, leading to potentially incorrect conclusions. In addition, the detection rate increases steeply with improved detector sensitivity, and it will be difficult to extract all the information from growing datasets. tiwari2021vamana move away from this approach by proposing a mixture model of weighted Gaussians (and a power law) that is expected to be capable of modeling a variety of complex distributions.
In this work, we continue this direction, but use normalizing flows tabak2010density ; rezende2015variational ; papamakarios2021normalizing as our model class. The main advantage is that normalizing flows have successfully been able to represent complex data distributions in previous works and can be fit to such distributions in a straightforward manner. We thus rid ourselves almost entirely of the (potentially biased) constraints put previously on the inference models.
Summarizing, we show that:

In the Hierarchical Bayesian Analysis (HBA) framework (a Bayesian graphical model used in the gravitational wave community), a normalizing flow can, despite having access to minimal and noisy data, recover structure in the data that agrees with either ground truth or robust results from previous studies. Further, it can reveal patterns that are easily missed by custom phenomenological modeling.

Using a prior over the free parameters of the normalizing flow, we can get a posterior predictive distribution that generates interpretable confidence intervals.
2 Method
Let be a gravitational wave time series and be an observed dataset of such events. We understand the physical implication of an event
through a vector of associated parameters
. For binary mergers, these parameters include both masses, the redshift, and effective spin. Parameter inference for individual events comes with a lot of uncertainty due to measurement noise intrinsic to the underlying time series. Further, these posterior distributions can have a nontrivial shape. This is exemplified in fig. 1 and fig. 2, where we show on the left the density estimates of
associated with for every. It is clear that heuristics such as taking the mean of the samples of the posterior distributions and fitting the population model to those is not sufficient. Therefore, the gravitational wave community has resorted to
Hierarchical Bayesian Analysis (HBA) mandel2019extracting ; gaebel2019digging ; vitale2022inferring , a Bayesian graphical model (see appendix A) for populationlevel inference. By taking into account measurement uncertainty, HBA distills the posterior samples into a single population model, which can be seen as a form of deconvolution bovy2011extreme . In this way, despite huge uncertainties in the parameters of individual events, we can still draw scientifically sound downstream conclusions about important characteristics of the Universe.2.1 Hierarchical Bayesian Analysis
We are interested in obtaining a population model for after observing . First, we introduce a model parameter with a fixed prior distribution . For a fully specified generative model, we further need to introduce distributions and , . We are interested in the posterior distribution
and the posterior predictive distribution
(1) 
which expresses the probability of observing a new event with parameters
given the (old) observations while taking into account the uncertainty in .While we usually do not have access to the marginal we can make use of a (usually physicsinformed prior) distribution , and then receive samples from the conditional , where . I.e., we assume that we have access to samples from the individual posterior distributions. As such, our dataset is where for every event we have samples from . We are first interested in maximizing an approximate posterior using :
(2) 
with normalizing constant (derivation appendix B). A similar MonteCarlobased estimator is presented in vandegar2021neural . After obtaining
through, e.g., stochastic gradient descent, we run Markov chain MonteCarlo around the MAP solution to obtain samples
. These samples and our model can be used to approximate the posterior predictive distribution eq. 1. It is important to model the whole posterior distribution, as the uncertainty in the parameters tells us where we can draw robust conclusions about the underlying population.2.2 Normalizing Flows
So far, we have not discussed how we parameterize . To facilitate increasingly larger datasets and to be able to model higherdimensional distributions, the goal of this research is to explore more flexible and scalable models. As such, we propose normalizing flows (tabak2010density, ; rezende2015variational, ; papamakarios2021normalizing, , appendix C) as a suitable model class. Normalizing flows transform a density using a chain of a smooth, invertible mappings such that which constructs an arbitrarily complex density. We parameterize each using a set of weights , and construct a loglikelihood which we can optimize for using, e.g., stochastic gradient descent. In this work, we use planar flows rezende2015variational and block neural autoregressive flows de2020block , as we experimentally found that they are sufficiently flexible while not using too many parameters. Furthermore, we found that the resulting distributions are smooth, which makes them physically more plausible and allows for better automated discovery of interpretable models wong2022automated .
3 Results
3.1 Illustrative Examples
In an illustrative example, we use a mixture of Gaussians as the ground truth marginal distribution over . The observable distribution is a Gaussian with mean , from which we draw . Using a uniform prior , we obtain posterior draws for all events using rejection sampling. Due to the noninvertible forward model, the posteriors are bimodal and noisy. Since the true underlying population distribution and the posteriors are nontrivial, standard techniques will not be effective here. Still, the normalizing flow can, using the Hierarchical Bayesian Analysis framework, convert the noisy posterior draws shown in fig. 1 (left) into a faithful population model as shown in fig. 1 (right). Further, we perform a more realistic (though less clearly visualizable) experiment with synthetic data in section D.2 where we sample data from previously suggested gravitational wave population models (e.g., by abbott2021population3 ) and recover the proposed distributions.
3.2 GWTC3: Primary Mass Marginal
In our experiments on the GWTC3 catalog, we consider four gravitional wave parameters: (primary mass), (secondary mass), (redshift) and (effective inspiral spin). I.e., and . Note that the number of recorded gravitational wave events is currently rather small (). Still, we show that we can retrieve structure from the data that agrees with previous studies. After getting by maximizing eq. 2, we obtain, using Hamiltonian Monte Carlo neal2011mcmc , samples , starting the chain at . Next, we obtain the posterior predictive distribution by the approximation . We show the resulting observed population posterior predictive marginal in fig. 2.
Note, again, that we did not impose any strong restrictions on the model class; this result was fully recovered from just the data. Overall, we see (apart from selection bias effects mandel2019extracting ) similarity with, e.g., the Power Law + Peak and MultiPeak model of abbott2021population ; abbott2021population3 . Specifically, we observe distinct peaks around , , and . Finally, we also reassuringly observe that the uncertainty contracts or expands in the parts where we expect it to. I.e., it expands at regions where we do not have many observations, and contracts in the regions where we do (e.g., around the highmass modes).
3.3 GWTC3: Observed Population Distribution
We display the marginal joint distributions in
fig. 3. Here, we define as the ratio of secondary to primary mass of the binary. Note this figure shows the observed population results. This is especially noticeable in the redshift marginal , which is heavily biased towards lowredshift events. Further, we expect more lower primary mass events . and are expected to be less affected by not including the selection function. To draw conclusions about the true population, one should include selection bias effects. Optimization was highly unstable when including the selection function in the objective. Hence, including selection bias effects was left for future investigation. Also, is expected to peak at , with an infinitely steep cutoff at . We did not observe this due to the smoothness of the flow. Future work could look into reparameterizations to enable this behavior.We can tentatively draw a few conclusions from the observed population figure. Interestingly, it suggests that and do not simply correlate through a power law. Instead, the distribution shows an inverted V shape. The redshift to mass correlations seem plausible and consistent. The effective spin marginal also aligns with previous literature. In mckernan2022ligo is was noted that does not correlate positively with , which is also confirmed here.
4 Discussion
We parameterized the population model in the Hierarchical Bayesian Analysis framework with a normalizing flow. In illustrative experiments, we confirmed that the model can retrieve the actual population density under nontrivial posterior samples. On gravitational wave data from GWTC3, we considered primary mass, secondary mass, redshift, and effective spin and recovered an observed population model that agrees with robust previous phenomenological modeling results despite the dataset being small and highly noisy. This paves the way for less constrained, tractable, and flexible populationlevel inference in noisy settings, especially once more higherdimensional data become available, allowing for automated discovery of structure in the data.
5 Broader Impact
This research proposed to combine normalizing flows with the HBA framework. The current work found an application to gravitational wave event data, but in principle, the method can be implemented in many fields of science. While this work purely aims to aid scientific discovery, the approach can also be used in nonscientific settings dealing with noisy observations, which can pose privacyrelated issues depending on the data type and source. As such, we encourage more data privacy awareness and establishing data protection policies.
6 Acknowledgements
This work was largely performed as part of the machine learning summer program of the Flatiron Institute. We would like to thank the scientific software development community, without whom this work would not be possible
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Appendix A Graphical Model
The graphical we use is shown in fig. 4. We colored the arrows that our normalizing flow parameterizes red. To estimate , samples are inferred for all in the first stage. In this work we assume we have those samples and use them to estimate the remaining densities in eq. 1 through the objective eq. 2.
Appendix B Hierarchical Bayesian Likelihood
Using , we set
(3)  
(4) 
Note that is a constant with respect to our model parameters . Hence,
(5)  
(6) 
Appendix C Normalizing Flows
To facilitate increasingly larger datasets and to be able to model higherdimensional distributions, the goal of this research is to explore more flexible and scalable models. As such, we propose normalizing flows[41, 37, 33] as a suitable model class. Normalizing flows transform a density using a smooth, invertible mapping with inverse
. As such, if we transform a random variable
, then has(7) 
If we use a chain of these transformations, such that
(8) 
we can construct an arbitrarily complex density. We parameterize each using a set of weights , and construct a loglikelihood
(9) 
which we can optimize for using, e.g., stochastic gradient descent. The downside is a potentially computationally expensive Jacobian determinant. As such, much work has been conducted on finding transformations that have computationally cheap Jacobian determinants [13, 25, 34, 6, 24, 19]. In this work, we use planar flows [37] and block neural autoregressive flows [12], as we experimentally found that they are sufficiently flexible while not using too many parameters. Furthermore, we found that the resulting distributions are (relative to some other parameterizations) smooth, which makes them physically more plausible.
Planar Flow
Planar flow uses the transformation
(10) 
where are free parameters, and is the hyperbolic tangent. These transformations contract and expand the inputs perpendicular to the plane defined . It can be shown that the Jacobian determinant of this mapping can be computed in [37]. As such, we can use a sequence of these transformations to model , our gravitational wave population model (eq. 2).
Block Neural Autoregressive Flow
Neural autoregressive flows decomposes a joint distribution over into conditional distributions. the transformations then yield lower triangular Jacobians and hence cheaply computable determinants. Block neural autoregressive flows then parameterize using a neural network that uses block lowertriangular matrices. Their diagonal elements are strictly positive, ensuring monotonicity. The offdiagonal elements do not require monotonicity as they only play a role in the conditioning part of the conditional distributions.
Appendix D Experimental Details
All experiments were conducted locally.
d.1 Illustrative Example
The groundtruth population is defined by a mixture . The forward model is given by . This noninvertible forward mapping results in noisy, bimodal posteriors. We sample events. Using a uniform prior and the true forward model, we draw posterior samples using rejection sampling from for every , . We parameterize as a 4layer planar flow and a standard Gaussian base distribution. We use the Adam optimizer [23] with learning rate so obtain . We found empirically that optimization runs better when minibatches are taken for both the events and also the posterior draws. In this experiment, we batch sizes of events and .
d.2 Synthetic Data Experiment
Here, we simulate a population model using previously published models and investigate whether our flowbased approach can faithfully recover those distributions. We start with the Power Law + Peak model for from [4] using the means of the posterior distributions for groundtruth population parameters. For a conditional effective spin distribution we use [10], again with the mean of the inferred distributions as groundtruth parameter values. Finally, we model the distribution of redshift using a power law with spectral index following [15]. Note that there is an extra factor in the merger coming from the comoving volume factor. The resulting samples are shown in fig. 5 (left). We simulate Gaussian posteriors in our Hierarchical Bayesian framework. The distribution that the normalizing flow recovers is shown on the right in fig. 5. We see that the flow recovers most of the correlations and shapes of the joint distribution faithfully.
d.3 Gwtc3
We have gravitational wave events available. The number of posterior samples per event ranges from to . We use 10 layers of planar flow, resulting in a total of free parameters, limiting the risk of severe overfitting resulting in unnatural population distribution shapes. We use minibatches of events and posterior samples per event. We use the Adam optimizer [23] with learning rate 0.001. to find .
Appendix E Future Work
In this study, we did not include selection bias
effects. Future work should adjust for this to obtain more realistic models of population parameters of gravitational waves. We also expect that we did not fully explore all parameterizations (and hyperparameters) that recover the population model as faithfully as possible, both in terms of the normalizing flow and how the input data should be transformed for the flow to optimally learn the densities.
Further, [49] have presented a promising approach to distill blackbox models into symbolic equations. Such an approach can also be applied to our normalizing flows, yielding more interpretable versions of the presented population densities.