Tutorials

Statistical inference refers to any method that tries to infer knowledge on an underlying distribution, e.g. neutron star masses, from analysis of a limited dataset, e.g. observed neutron stars. In the specific case of Bayesian Inference this is done by applying Bayes' theorem. The sought after posterior distribution results from reweighting a certain priordistribution by how well it satisfies certain data constraints that are expressed by a likelihood function.

In the context of astrophysical parameter estimation, the prior and posterior may refer to the parameter space of compact systems, comprising masses, kinetic and angular quantities and many more. These distributions are usually continuous quantities and thus need to be discretised for numerical treatment. As the number of parameters increases, the numeric cost quickly becomes prohibitive even for a very coarse discritisation of the parameter space. Bilby, the Bayesian inference library provides convenient routines to evade this problem by implementing the ideas of Nested Sampling tailored to the needs of compact binary coalescence research.

The fundamental idea is that for any meaningful inference, the posterior should peak in a much narrower region of parameter space than the prior. Instead of drawing samples that cover prior space uniformly, nested sampling algorithms draw samples from nested shells with increasing likelihood that naturally contract around the posterior distrubition's modes.

To generate an injection file, NMMA provides the command-line utility nmma_create_injection. This generates a json style injection file by drawing from a prior file. An example of a prior file called bns.prior for binary neutron star systems with aligned spins is given below:

mass_1 = Uniform(name='mass_1', minimum=1., maximum=2.)
mass_2 = Uniform(name='mass_2', minimum=1., maximum=2.)
chi_1 = bilby.gw.prior.AlignedSpin(name='chi_1', a_prior=Uniform(minimum=0, maximum=0.05))
chi_2 = bilby.gw.prior.AlignedSpin(name='chi_2', a_prior=Uniform(minimum=0, maximum=0.05))
luminosity_distance = bilby.gw.prior.UniformComovingVolume(name='luminosity_distance', minimum=10, maximum=100, unit='Mpc', latex_label='$d_L$')
dec =  Cosine(name='dec', boundary='reflective')
ra =  Uniform(name='ra', minimum=0, maximum=2 * np.pi, boundary='periodic')
cos_theta_jn = Uniform(name='cos_theta_jn', minimum=-1, maximum=1)
psi = Uniform(name='psi', minimum=0, maximum=np.pi, boundary='periodic')
phase = Uniform(name='phase', minimum=0, maximum=2 * np.pi, boundary='periodic')

Running the command

nmma_create_injection --prior-file bns.prior -n 10 -s 42 --eos-file <EOSdata.dat> --binary-type BNS -e json -f injection_bns --duration 128

will create an injection file injection_bns.json for 10 BNS sources each having a duration of 128 s. The eos-file command is useful, when information from a particular EOS should be used. Here, the path and file name to the EOS file needs to be provided.

Taking the Metzger (2017) blue kilonova model as an example, a set of injections can be generated by simply using this Me2017.prior file (all are found in nmma/priors/). Example EOS files can be found here. Using the same command as above

nmma_create_injection --prior-file priors/Me2017.prior --eos-file example_files/eos/ALF2.dat --binary-type BNS --n-injection 100 --original-parameters --extension json

an injection.json is generated with the Me2017.prior including a number of injections specified by –n-injection.