Parameter Estimation with Bilby

Given an interpolation table of the lensing amplification factor and an unlensed waveform model, the lensed waveform can be rapidly called.

While any value on the \(y\)-\(l\) plane (where \(y\) is the impact parameter and \(l\) the lens parameter) can be interpolated in principle as long as the value lies within the boundaries of the interpolation table, the interpolated amplification is ill-behaved close to the caustic because the magnification diverges formally. Therefore, when sampling over the \(y\)-\(l\) plane for parameter estimation, we will ignore the region close to the caustic by assigning it a zero prior probability as implemented in the glworia.pe.prior module. We define such a region as the region between \(y_c(l) \pm \delta\), where \(y_c(l)\) is the caustic curve in the \(y\)-\(l\) plane, and \(\delta\) is a constant. For lens models that will reduce to the SIS lens at certain limits (e.g. \(x_c \to 0\) for the CIS lens, or \(k \to 1\) for gSIS), we also exclude the region close to the \(y = 1\) caustic at those limits by setting the prior probability within a small rectangle to be zero.

An example parameter estimation script, scripts/PE_full.py, can be found in the git repository. The scripts can be run with a configuration file.

python scripts/PE_full.py /path/to/config_file.json

Example configuration files can be found in the config/ directory. The dictionaries required in the json configuration file are as follows:

• injection_parameters: the injected parameters for the lensed waveform. These include all the parameters of the unlensed waveform (passed to the waveform generator in bilby), in addition to the following lens parameters:

  Name	Type	Description
  y	float	Injected impact parameter for the lensing set-up
  MLz	float	Injected redshifted lens mass
  lp	float	Injected lens parameter of the lens

• waveform_arguments: the keyword arguments to be passed to the bilby.gw.waveform_generator.WaveformGenerator object in bilby.

• sampler_settings: the keyword arguments to be passed to the bilby.run_sampler function.
• interpolator_settings: the settings of the interpolator used, see the tutorial for loading the interpolation tables for more details.

• prior_settings: the settings for the priors of the lensing related parameters and luminosity_distance. For example, the prior type (uniform, loguniform) and limits of MLz can be specified by MLz_prior_type, MLz_min and MLz_max. For luminosity_distance, the prior can be chosen to be uniformsourceframe. Additional arguements for masking the region near the caustic are as follows:

  Name	Type	Description
  crit_mask_settings	dict	Parameters of the masker around the caustic, including fac, cap_high and cap_low. The masked region is defined to be between y_c - min( y_c * fac, y_c - cap_low ) and y_c + min( y_c * fac, y_c + cap_high).
  mask_boxes	list	Each item is a list [[y_low, y_high],[l_low, l_high]], the boundaries of a rectangular region to exclude in the \(y\)-\(l\) plane

• misc: miscellaneous settings as follows:

  Name	Type	Description
  zero_noise	bool	Whether to perform a zero-noise injection.
  sampling_frequency	float	The sampling rate of the signal in Hertz.
  minimum_frequency	float	Minimum frequency cut-off.
  seed	int	The seed used to generate random numbers for sampling.
  lp_name	str	The name given to the lens parameter, e.g. kappa for the NFW lens.
  lp_latex	str	The name of the lens parameter in latex form for making plots
  outdir_ext	str	The name of the output directory