Bilby: A User-friendly Bayesian Inference Library for Gravitational-wave Astronomy

Bilby is open-source, MIT-licensed, and written in Python. The simplest installation method is through PyPI.¹⁵ The following command installs from the command line:

$$\begin{eqnarray*}&&\$\ {\mathtt{pip}}\ {\mathtt{install}}\ {\mathtt{bilby}}.\end{eqnarray*}$$

This command downloads and installs the package and dependencies. The source code can be obtained from the git repository,¹⁶ which also houses an issue tracker and merge-request tool for those wishing to contribute to code development. This repository is mirrored on github¹⁷ and is archived in Zenodo (doi:10.5281/zenodo.2602178). Documentation about code installation, functionality, and user syntax is also provided.¹⁸ Scripts to run all examples presented in this work are provided in the git repository.

Bilby has been designed such that logical blocks of code are separated, and, wherever possible, code is abstracted away to allow future reuse by other models. At the top level, Bilby has three packages: core, gw, and hyper. The core package contains the key functionalities. It passes the user-defined priors and likelihood function to a sampler, harvests the posterior samples and evidence calculated by the sampler, and returns a result object providing a common interface to the output of any sampler along with information about the inputs. The gw package contains gravitational-wave-specific functionality, including waveform models, gravitational-wave-specific priors, and likelihoods. The hyper package contains functionality for the hierarchical Bayesian inference (see Section 6). A flowchart showing the dependency of different packages and modules is available on the git repository.¹⁹

3.2.1. The core Package

3.2.2. The gw Package

The hyper package contains all required functionality to perform hierarchical Bayesian inference of populations. This includes both a Model module and a HyperparameterLikelihood class. This entire package is discussed in more detail in Section 6.

The first direct detection of gravitational waves occurred on the 2015 September 14, when the two LIGO detectors (Aasi et al. 2015) in Hanford, Washington, and Livingston, Louisiana, detected the coalescence of a binary black hole system (Abbott et al. 2016d). The gravitational waves swept through the two detectors with a ${6.9}_{-0.4}^{+0.5}$ ms time difference that, when combined with polarization information, allowed for a sky location reconstruction covering an annulus of 590 deg² (Abbott et al. 2016d). The initially published masses of the colliding black holes were given as ${36}_{-4}^{+5}$ and ${29}_{-4}^{+4}$ (Abbott et al. 2016e). Subsequent analyses with more accurate precessing waveforms constrained the masses to be ${35}_{-3}^{+5}$ and ${30}_{-4}^{+3}$ at 90% confidence (Abbott et al. 2016c). The distance to the source is determined to be ${440}_{-180}^{+160}$ Mpc (Abbott et al. 2016c).

In this example, we use Bilby to reproduce the parameter estimation results for GW150914. The data for published LIGO/Virgo events are made available through the Gravitational Wave Open Science Center (Vallisneri et al. 2015). Built-in Bilby functionality downloads and parses this data. We begin with the following two lines:

$$\begin{eqnarray*}&&\begin{array}{l}\gt \gt \gt \,{\mathtt{importbilby}}\\ \gt \gt \gt \,{\mathtt{interferometers}}\,=\,{\mathtt{bilby}}.{\mathtt{gw}}.{\mathtt{detector}}.{\mathtt{get}}\_{\mathtt{event}}\_{\mathtt{data}}(^{\prime\prime} {\mathtt{GW}}{\mathtt{150914}}^{\prime\prime} )\end{array}.\end{eqnarray*}$$

The first line of code imports the Bilby code base into the Python environment. The second line returns a set of objects that contain the relevant data segments and associated data products relevant for the analysis for both the LIGO Hanford and Livingston detectors. By default, Bilby downloads and windows the data. A local copy of the data is saved, along with diagnostic plots of the gravitational-wave strain amplitude spectral density.

In addition to the data, the two key ingredients for any Bayesian inference calculation are the likelihood and the prior. Default sets of priors can be called from the gw.prior module, and we also employ the default Gaussian noise likelihood (Equation (1)):

$$\begin{eqnarray*}&&\begin{array}{l}\gt \gt \gt \,{\mathtt{prior}}={\mathtt{bilby}}.{\mathtt{gw}}.{\mathtt{prior}}.{\mathtt{BBHPriorDict}}({\mathtt{filename}}\,=\,^{\prime\prime} {\mathtt{GW}}{\mathtt{150914}}.{\mathtt{prior}}^{\prime\prime} )\\ \gt \gt \gt \,{\mathtt{likelihood}}\,=\,{\mathtt{bilby}}.{\mathtt{gw}}.{\mathtt{likelihood}}.{\mathtt{get}}\_{\mathtt{binary}}\_{\mathtt{black}}\_{\mathtt{hole}}\_{\mathtt{likelihood}}({\mathtt{interferometers}})\end{array}.\end{eqnarray*}$$

The above code calls the GW150914 prior, which differs from the priors described in Table 1 in two main ways. First, to speed up the running of the code, it restricts the mass priors to between 30 and 50 M_⊙ for the primary mass and 20 and 40 M_⊙ for the secondary mass. Moreover, this prior call restricts the time of coalescence to 0.1 s before and after the known coalescence time. One can revert to the priors in Table 1 by replacing the above file call with filename=''binary_black_holes.prior'', but this would require separately setting a prior for the coalescence time. We show how this can be done in Section 4.2.

The next step is to call the sampler:

$$\begin{eqnarray*}&&\begin{array}{l}\gt \gt \gt {\mathtt{result}}\,=\,{\mathtt{bilby}}.{\mathtt{core}}.{\mathtt{sampler}}.{\mathtt{run}}\_{\mathtt{sampler}}({\mathtt{likelihood}},{\mathtt{prior}})\end{array}.\end{eqnarray*}$$

This line performs parameter inference using the sampler default Dynesty, with a default 500 live points. This number can be increased by passing the nlive= keyword argument to run_sampler(). The sampler returns a list of posterior samples, the Bayesian evidence, and metadata, which are stored in an hdf5 file. One may plot a corner plot showing the posterior distribution for all parameters in the model using the command

$$\begin{eqnarray*}&&\gt \gt \gt {\mathtt{result}}.{\mathtt{plot}}\_{\mathtt{corner}}().\end{eqnarray*}$$

The above example code produces posterior distributions that, by eye, agree reasonably well with the parameter uncertainty associated with the published distributions for GW150914. The shape of the likelihood for the extrinsic parameters presents significant challenges for samplers due to strong degeneracies between different sky locations, distances, inclination angles, and polarization angles (see, e.g., Farr et al. 2014a; Raymond & Farr 2014). For more accurate results, we use the nested sampling package CPNest (Veitch et al. 2017), which is invoked by changing the run_sampler function above to include the additional argument sampler='cpnest'. We also change the number of live points by adding nlive=5000 to the same function and specify a keyword argument, maxmcmc=5000, which is the maximum number of steps the sampler takes before accepting a new sample. To resolve the issue with the phase at coalescence, we analytically marginalize over this parameter (Farr 2014) by adding the optional phase_marginalization=True argument to the instantiation of the likelihood. Bilby has built-in analytic marginalization procedures for the time of coalescence (Farr 2014) and distance (Singer & Price 2016; Singer et al. 2016), which can both be invoked using time_marginalization=True and distance_marginalization=True, respectively. These decrease the run time of the code by minimizing the dimensionality of the parameter space. Posterior distributions can still be determined for these parameters by reconstructing them analytically from the full set of posterior samples (e.g., see Thrane & Talbot 2018).

Using Bilby, we can plot marginalized distributions by simply passing the plot_corner function the optional parameters=... argument. In Figure 1, we show the marginalized, two-dimensional posterior distribution for the masses of the two black holes as calculated using the above Bilby code (shown in blue). In orange, we show the LIGO posterior distributions from Abbott et al. (2016a), calculated using the LALInference software (Veitch et al. 2015) and hosted at the Gravitational Wave Open Science Center (Vallisneri et al. 2015).

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In Figure 2, we show the marginalized posterior distribution of the luminosity distance and inclination angle, where the Bilby posteriors are again shown in blue and the LALInference posteriors in orange. Figure 3 shows the sky localization uncertainty for both Bilby and LALInference.

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The above example does not make use of detector calibration uncertainty, which is an important feature in LIGO data analysis. Such calibration uncertainty is built into Bilby using the cubic spline parameterization (Farr et al. 2014b), with example usage in the Bilby repository.

Bilby supports both the analysis of real data, as in the previous section, and the ability to inject simulated signals into Monte Carlo data. In the following two sections, we inject a binary black hole signal and a binary neutron star signal, respectively, showing how one can easily inject and recover signals and their astrophysical properties.

In this first example,²⁶ we create a binary black hole signal with parameters similar to GW150914 (Abbott et al. 2016e), albeit at a luminosity distance of d_L = 2 Gpc (see d_L ≈ 400 Mpc for GW150914). We inject the signal into a network of LIGO-Livingston, LIGO-Hanford (Aasi et al. 2015), and Virgo interferometers (Acernese et al. 2015), each operating at design sensitivity. When doing examples of this nature, it is time-intensive to sample over all 15 parameters in the waveform model. Therefore, to get quick results that can be run on a laptop, we only sample over four parameters in the waveform model: the two black hole masses m_1,2, the luminosity distance d_L, and the inclination angle $\iota$. Bilby supports simple functionality to limit or extend the number of parameters included in the likelihood calculation, as shown below.

We begin by setting up a WaveformGenerator object using a frequency domain strain model that takes the signal injection parameters and specific waveform arguments, such as the waveform approximant, as arguments. The WaveformGenerator also takes data duration and sampling frequency as input parameters. With the source model defined, we now instantiate an interferometer object that takes the strain signal from the WaveformGenerator and injects it into a noise realization of the three interferometers. One could choose to do a zero-noise simulation by simply including the flag zero_noise=True.

Priors are set up as in the previous open-data example, except we call the binary_black_holes.prior file instead of the specific prior file for GW150914. Moreover, to hold all but four of the parameters fixed, we set the value of the prior for those other parameters to the injection value. For example, setting

$$\begin{eqnarray*}&&\gt \gt \gt {\mathtt{prior}}[^{\prime} {\mathtt{a}}\_{\mathtt{1}}^{\prime} ]={\mathtt{0}}\end{eqnarray*}$$

sets the prior on the dimensionless spin magnitude of the primary black hole to a δ-function at zero.

In general, we can change the prior for any parameter with one line of code. For example, to change the prior on the primary mass to be uniform between m₁ = 25 and 35 M_⊙, say, one includes

$$\begin{eqnarray*}&&\begin{array}{l}\gt \gt \gt {\mathtt{prior}}[^{\prime} {\mathtt{mass}}\_{\mathtt{1}}^{\prime} ]\,=\,{\mathtt{bilby}}.{\mathtt{core}}.{\mathtt{prior}}.{\mathtt{Uniform}}({\mathtt{minimum}}={\mathtt{25}},{\mathtt{maximum}}={\mathtt{35}},{\mathtt{unit}}={\mathtt{r}}^{\prime} \${\mathtt{M}}\_\setminus {\mathtt{odot}}\$^{\prime} )\end{array}.\end{eqnarray*}$$

Bilby knows about many different types of priors that can all be called in this way.

For this example, we are also required to define priors on the coalescence time, which we define to be a uniform prior with minimum and maximum 1 s either side of the injection time.

The likelihood is again set up similarly to the open-data example of Section 4.1, although this time, we must pass the interferometer, waveform_generator, and prior. Finally, the sampler can be called in the same way as Section 4.1; for this example, we use the pyMultiNest nested sampler (Buchner et al. 2014).

Figure 4 shows the recovered posterior distributions (blue) and injected parameter values (orange). For this example, using the PyMultiNest (Buchner et al. 2014) nested sampling package with 6000 live points took approximately 30 minutes on a laptop to fully sample the four-dimensional parameter space. The parameters in Figure 4 are recovered well with the usual degeneracy present between the luminosity distance and inclination angle of the source, d_L and $\iota$, respectively.

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The first detection of binary neutron star coalescence GW170817 was a landmark event signaling the beginning of multimessenger gravitational-wave astronomy (Abbott et al. 2017g, 2017h). Gravitational-wave parameter estimation of the inspiral is what ultimately determined that both objects were likely neutron stars and provides the best-yet constraints on the nuclear equation of state of matter at supranuclear densities (Abbott et al. 2017c, 2017g, 2017h).

One of the key measurements in determining the equation of state from binary neutron star coalescences is that of the tidal parameters. The dimensionless tidal deformability

$$\begin{eqnarray}&&{\rm{\Lambda }}=\displaystyle \frac{2{k}_{2}}{3}{\left(\displaystyle \frac{{c}^{2}R}{Gm}\right)}^{5}\end{eqnarray} \tag{ 2 }$$

is a fixed parameter for a given equation of state and neutron star mass. Here k₂ is the second Love number, and R and m are the neutron star radius and mass, respectively. The binary neutron star merger GW170817 provided constraints of ${{\rm{\Lambda }}}_{1.4}={190}_{-120}^{+390}$ (Abbott et al. 2018b; De et al. 2018), where the subscript denotes that this is the estimate on Λ assuming a 1.4 M_⊙ neutron star, and the uncertainty is the 90% credible interval.

Bilby can be used to study neutron star coalescences in both real and simulated data. We inject a binary neutron star signal using the TaylorF2 waveform approximant into a three-detector network of the two LIGO detectors and Virgo, all operating at design sensitivity.²⁷ Our injected signal is an ${m}_{1}=1.3\,{M}_{\odot },{m}_{2}=1.5\,{M}_{\odot }$ binary at d_L = 50 Mpc with dimensionless spin parameters a_1,2 = 0.02 and tidal deformabilities Λ_1,2 = 400. Setting up such a system in Bilby is equivalent to doing the binary black hole injection study of Section 4.2, except we call the lal_binary_neutron_star source function, which requires the additional Λ_1,2 arguments. We also have specific binary neutron star priors; the default set can be called using

$$\begin{eqnarray*}&&\gt \gt \gt \,{\mathtt{priors}}={\mathtt{bilby}}.{\mathtt{gw}}.{\mathtt{prior}}.{\mathtt{BNSPriorDict}}().\end{eqnarray*}$$

The standard set of binary neutron star priors is shown in Table 2. In this example, we use the Dynesty sampler.

Table 2. Default Binary Neutron Star Priors

Variable	Unit	Prior	Minimum	Maximum
m1,2	M⊙	Uniform	1	2
a1,2	⋯	Uniform	−0.05	0.05
Λ1,2	⋯	Uniform	0	3000
dL	Mpc	Comoving	10	500
R.A.	rad.	Uniform	0	2π
Decl.	rad.	cos	−π/2	π/2
$\iota$	rad.	sin	0	π
ψ	rad.	Uniform	0	π
ϕc	rad.	Uniform	0	2π

Note. Here Λ_1,2 are the tidal deformability parameters of the primary and secondary neutron star defined in Equation (2). For other variable definitions, see Table 1. Note that our commonly used waveform approximant does not allow misaligned neutron star spins, implying that we do not require priors on those spin parameters.

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The tidal deformability parameters Λ₁ and Λ₂ are known to be highly correlated. The terms that appear explicitly due to the tidal corrections in the phase evolution are instead $\tilde{{\rm{\Lambda }}}$ and $\delta \tilde{{\rm{\Lambda }}}$ (Flanagan & Hinderer 2008; for definitions of these parameters, see Equations (14) and (15) of Lackey & Wade 2015). We therefore sample in $\tilde{{\rm{\Lambda }}}$ and $\delta \tilde{{\rm{\Lambda }}}$ instead of Λ₁ and Λ₂. Although we sample in all binary neutron star parameters, we show only the two-dimensional marginalized posterior distribution for $\tilde{{\rm{\Lambda }}}$ and $\delta \tilde{{\rm{\Lambda }}}$ in Figure 5. The corresponding injected values of $\tilde{{\rm{\Lambda }}}$ and $\delta \tilde{{\rm{\Lambda }}}$ are shown as the orange vertical and horizontal lines, respectively.

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The preceding subsections have only given the flavor of what can be achieved with Bilby for compact binary coalescences. It is trivial to implement more complex signal models that include, for example, higher-order modes, eccentricity, gravitational-wave memory, nonstandard polarizations. Examples showing different signal models are included in the git repository.²⁸ Bilby has already been used in one such application: testing how well the orbital eccentricity of binary black hole systems can be measured with Advanced LIGO and Advanced Virgo (Lower et al. 2018). An example script reproducing those results can be found in the git repository.

If a signal model exists in the LAL software,²⁹ then calling that signal model and defining which parameters to include in the sampler is as simple as the above examples. In Section 5, we also show how to include a user-defined source model. Moreover, one is free to define and sample models in either the time or frequency domain. We include examples for both cases in the git repository. The latter case of using a time-domain source model requires doing little more than selecting the argument time_domain_source_model in the WaveformGenerator, rather than selecting frequency_domain_source_model.

Of course, one may also want to set up the injection and sampler using two different waveform models, for example, to inject a numerical relativity signal into Monte Carlo data and recover it with a waveform approximant (see also Section 5.1). This is possible by simply instantiating two WaveformGenerators, injecting with one and passing the other to the likelihood.

The full network of ground-based gravitational-wave interferometers will soon consist of the two LIGO detectors in the US, Virgo, LIGO-India (Iyer et al. 2011), and the KAGRA detector in Japan (Aso et al. 2013), all of which are implemented in Bilby. A gravitational-wave interferometer is specified by its geographic coordinates, orientation, and noise power spectral density. By default, Bilby includes descriptions of current detectors, including LIGO, Virgo, and KAGRA, as well as proposed future detectors A+ (Miller et al. 2015), Cosmic Explorer (Abbott et al. 2017b), and the Einstein Telescope (Punturo et al. 2010). It is also possible to define new detectors, which is useful for developing the science case for proposals and to optimize the design and placement of new detectors. Among other things, this can be used in developing the science case for interferometer design and placement.

Bilby provides a common interface to define detectors by their geometry, location, and frequency response. By way of example, we place a new 4 km arm interferometer in the Shire of Gingin, located outside of Perth, Australia, the current location of the Australian International Gravitational Observatory. We assume a futuristic network configuration of the Australian Observatory, together with the two LIGO detectors in Hanford and Livingston, all operating at A+ sensitivity (Miller et al. 2015). We generate A+ power spectral densities in the same script used to run Bilby by using the pygwinc software,³⁰ which creates an array containing the frequency and noise power spectral density³¹ (one could equally use more sophisticated software such as Finesse (Brown & Freise 2014) to create more detailed interferometer sensitivity curves). We then create a new Interferometer object using bilby.gw.detector.Interferometer(), which takes numerous arguments, including the position and orientation of the detector, minimum and maximum frequencies, and power or amplitude noise spectral density. The noise spectral density can be passed as an ascii file containing the frequency and spectral noise density. With the new detector defined, one can again calculate a noise realization and signal injection in a manner similar to what is done in Section 4.

In this example, we inject a GW150914-like binary black hole inspiral signal at a luminosity distance of d_L = 4 Gpc and recover the masses, sky location, luminosity distance, and inclination angle of the system. In this example, we use the Nestle sampler. Figure 6 shows the two-dimensional marginalized posterior for the sky location uncertainty when including (blue) and not including (orange) the Australian detector in Gingin. In this instance, the sky localization uncertainty decreases by approximately a factor of four when including the third detector.

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While this example includes three detectors, it is straightforward to extend this analysis to an arbitrary detector network. The likelihood evaluation simply loops over the number of detectors passed to it and multiplies the likelihood for each detector to get a combined likelihood for each point in the parameter space.

Gravitational-wave signals from core-collapse supernovae are complicated and not well understood in terms of their specific phase evolution. Numerous techniques have been developed to deal with both detection and parameter estimation. One such method for the latter problem involves principal component analysis (Logue et al. 2012; Powell et al. 2016, 2017), where the signal is reconstructed using a weighted sum of orthonormal basis vectors. In this example, we inject a gravitational-wave signal from a numerical relativity simulation (Müller et al. 2012) and recover the principal components using Bilby.³²

The injection is performed by defining a new signal class that, in this case, simply reads in an ascii text file containing the gravitational-wave strain time series. The injection is then performed in a way akin to the binary black hole and binary neutron star examples in Section 4. We inject signal L15 from Müller et al. (2012), which comes from a three-dimensional simulation of a nonrotating core-collapse supernova with a 15 M_⊙ progenitor star. The signal is injected at a distance of 5 kpc in the direction of the galactic center. The amplitude spectral density of the injected signal is shown in Figure 7 as the orange line.

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The signal is reconstructed using principal component analysis, such that the strain is expressed as

$$\begin{eqnarray}&&\tilde{h}(f)=A\sum _{j=1}^{k}{\beta }_{j}{U}_{j}(f),\end{eqnarray} \tag{ 3 }$$

where A is an amplitude factor, and β_j and U_j are the complex principal component amplitudes and vectors, respectively. Equation (3) is implemented into Bilby as another new signal model that takes the β_j coefficients, luminosity distance (which is a proxy for A), and sky location as inputs. Priors for each of the new parameters are established in the same way as the example with the mass in Section 4.2. In this case, we set k = 5 and use uniform priors between −1 and 1 for each of the β_j values.

Figure 7 shows the injected (orange) and recovered (blue) gravitational-wave signal in the frequency domain. The dark blue curve shows the maximum-likelihood curve, and the shaded blue region is a superposition of many reconstructed waveforms from the posterior samples.

There are a number of physical scenarios that can occur following the merger of two neutron stars, including the existence of short- or long-lived neutron star remnants. In the early phases post-merger (≲1 s), these neutron stars are highly dynamic and can emit significant gravitational radiation potentially observable by Advanced LIGO and Virgo at a design sensitivity out to ∼50 Mpc (e.g., Clark et al. 2014 and references therein). While the ultimate fate of binary merger GW170817 is unknown, no gravitational waves from a post-merger remnant were found (Abbott et al. 2017i, 2018c), which is not surprising given that the interferometers were not operating at design sensitivity and the distances involved.

Provided the sensitivity of gravitational-wave interferometers continues to increase, it is possible that a gravitational-wave signal from a post-merger remnant could be detected in the relatively near future. Such a detection would provide an excellent opportunity to understand the nuclear equation of state of matter at extreme densities, as well as the rich physics of these exotic objects (e.g., Shibata & Taniguchi 2006; Baiotti et al. 2008; Read et al. 2013). Parameter inference of such short-lived signals is in its infancy (e.g., see Chatziioannou et al. 2017), largely due to the paucity of reliable waveforms (Clark et al. 2016; Easter et al. 2018). This is an ongoing challenge due to the expensive nature of numerical relativity simulations and the complex physics that must be included in such simulations.

Simple models that provide approximate gravitational-wave signals fit to a handful of numerical relativity waveforms exist (Messenger et al. 2014; Bose et al. 2018; Easter et al. 2018), which may eventually be used for full parameter inference. The phase evolution of such numerical relativity simulations is rapid and very difficult to model (Messenger et al. 2014; Easter et al. 2018). However, it is the frequency content of the signal that carries information about the equation of state and the physics of the remnant (e.g., Takami et al. 2015 and references therein). It is therefore possible that parameter estimation algorithms may require one to throw away information about the phase and only keep amplitude spectral content. Such a process requires a different likelihood function than the one that has been used to this point. This, therefore, provides good motivation for showing how to include a different likelihood function in the Bilby code.

We implement a power–spectral density ("burst"') likelihood,

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}{ \mathcal L }(| d| \,| \,\theta ) & = & \displaystyle \sum _{i=1}^{N}\left[\mathrm{ln}{I}_{0}\left(\displaystyle \frac{| {\tilde{h}}_{i}(\theta )| | \tilde{{d}_{i}}| }{{S}_{n}({f}_{i})}\right)-\displaystyle \frac{| {\tilde{h}}_{i}{| }^{2}+| {\tilde{d}}_{i}{| }^{2}}{2{S}_{n}({f}_{i})}\right.\\ & & \left.+\,\mathrm{ln}| {\tilde{h}}_{i}(\theta )| -\mathrm{ln}{S}_{n}({f}_{i})\right],\end{array}\end{eqnarray} \tag{ 4 }$$

where I₀ is the zeroth-order modified Bessel function of the first kind. This requires setting up a new Likelihood class that contains a log_likelihood function that reads in the frequency array, noise spectral density, and waveform model and outputs a single likelihood evaluation. Having defined a new likelihood function, one calls the remaining functions in the usual way; the likelihood function is instantiated and passed to the run_sampler() command.

We inject a double-peaked Gaussian, shown in Figure 8 as the orange curve. We recover this signal using the same model (with a constant noise spectral density), where we use uniform priors for the amplitudes, widths, and frequencies of each of the peaks. Figure 8 shows the waveform reconstruction for each of the posterior samples, which can be seen to cover the injected signal.

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