Bayesian Inference

Pipeline A — self-contained MWE for Hubble constant inference (dev cross-check).

This module provides BayesianInference, a minimal working example that estimates the Hubble constant posterior from synthetic EMRI detections and a GalaxyCatalog. It uses:

  • An erf-based analytic GW detection probability,

  • A hardcoded 10 % fractional σ(d_L) (see FRACTIONAL_LUMINOSITY_ERROR),

  • A synthetic galaxy catalog generated by GalaxyCatalog.

This pipeline is NOT invoked by ``–evaluate``. It exists for rapid prototyping and cross-checking. The production (science-grade) pipeline is BayesianStatistics (Pipeline B), which uses the real GLADE galaxy catalog, KDE-based detection probability, full Fisher-matrix covariance, and multiprocessing.

The __main__ entry point in bayesian_inference_mwe runs this pipeline standalone.

class master_thesis_code.bayesian_inference.bayesian_inference.BayesianInference(galaxy_catalog, emri_detections, luminosity_distance_threshold=1.55, number_of_redshift_steps=1000, redshift_values=<factory>, galaxy_distribution_at_redshifts=<factory>, galaxy_detection_mass_distribution_at_redshifts=<factory>, detection_skylocalization_weight_by_galaxy=<factory>, use_bh_mass=False, use_selection_effects_correction=True)[source]

Bases: object

Bayesian estimator for the Hubble constant H₀ from EMRI detections.

Evaluates \(p(H_0 \mid \{d_i\})\) by marginalizing each detection over the galaxy-catalog redshift distribution, weighted by the GW likelihood and sky-localization overlap.

Variables:

  • galaxy_catalog (master_thesis_code.datamodels.galaxy.GalaxyCatalog) – Galaxy catalog providing redshift and mass distributions \(p(z)\) and \(p(M)\).

  • emri_detections (list[master_thesis_code.datamodels.emri_detection.EMRIDetection]) – List of EMRI detections with measured luminosity distances and sky positions.

  • luminosity_distance_threshold (float) – LISA detection horizon in Gpc.

  • number_of_redshift_steps (int) – Number of quadrature points for redshift integrals.

  • redshift_values (numpy.ndarray[tuple[Any, ...], numpy.dtype[numpy.float64]]) – Redshift grid used for all integrals, set in __post_init__.

  • galaxy_distribution_at_redshifts (numpy.ndarray[tuple[Any, ...], numpy.dtype[numpy.float64]]) – Pre-computed \(p(z_i \mid \mathrm{catalog})\) evaluated on redshift_values.

  • use_bh_mass (bool) – If True, include BH mass information in the likelihood.

  • use_selection_effects_correction (bool) – If True, divide likelihood by the selection-function denominator \(\int p_\mathrm{det}(z) p(z) dz\).

Parameters:
detection_skylocalization_weight_by_galaxy: list
emri_detections: list[EMRIDetection]
galaxy_catalog: GalaxyCatalog
galaxy_detection_mass_distribution_at_redshifts: list
galaxy_distribution_at_redshifts: ndarray[tuple[Any, ...], dtype[float64]]
gw_detection_probability(redshift, hubble_constant)[source]

Probability that a source at redshift is detected by LISA.

Models the detector selection function as a cumulative normal:

\[p_\mathrm{det}(z) = \frac{1}{2}\left[1 + \mathrm{erf}\! \left(\frac{D_\mathrm{thr} - d_L(z)}{\sqrt{2}\,\sigma_{d_L}(z)}\right)\right]\]

where \(\sigma_{d_L} = f_\sigma \cdot d_L(z)\) and \(f_\sigma\) is FRACTIONAL_LUMINOSITY_ERROR.

Parameters:

  • redshift (float) – Source redshift.

  • hubble_constant (float) – Dimensionless Hubble parameter \(h\).

Returns:

Detection probability in \([0, 1]\).

Return type:

float

gw_likelihood(measured_luminosity_distance, redshift, hubble_constant)[source]

GW likelihood: probability of measuring measured_luminosity_distance given redshift.

Assumes Gaussian measurement noise:

\[\mathcal{L}(\hat{d}_L \mid z, h) = \mathcal{N}\!\left(\hat{d}_L;\, d_L(z, h),\, f_\sigma \cdot d_L(z, h)\right)\]
Parameters:

  • measured_luminosity_distance (float) – Measured luminosity distance \(\hat{d}_L\) in Gpc.

  • redshift (float) – True source redshift.

  • hubble_constant (float) – Dimensionless Hubble parameter \(h\).

Returns:

Likelihood value (probability density).

Return type:

float

likelihood(hubble_constant, measured_luminosity_distance, measured_redshifted_mass, detection_index)[source]

Marginalized likelihood \(p(\hat{d}_L \mid H_0)\) for one EMRI detection.

Integrates the GW likelihood over the galaxy-catalog redshift distribution, weighted by sky-localization overlap and — optionally — BH mass information. A selection-effects correction divides by the denominator integral when use_selection_effects_correction is True:

\[\mathcal{L}(H_0) = \frac{\int p_\mathrm{GW}(\hat{d}_L \mid z, H_0)\, p_\mathrm{det}(z, H_0)\, p(z \mid \mathrm{catalog})\, dz} {\int p_\mathrm{det}(z, H_0)\, p(z \mid \mathrm{catalog})\, dz}\]
Parameters:

  • hubble_constant (float) – Dimensionless Hubble parameter \(h\) being evaluated.

  • measured_luminosity_distance (float) – Measured \(\hat{d}_L\) in Gpc.

  • measured_redshifted_mass (float) – Measured redshifted BH mass \(M_z\) in solar masses.

  • detection_index (int) – Index into emri_detections and pre-computed weight arrays.

Returns:

Likelihood value (positive real number; not normalized over \(H_0\)).

Return type:

float

luminosity_distance_threshold: float = 1.55
number_of_redshift_steps: int = 1000
posterior(hubble_constant)[source]
Parameters:

hubble_constant (float)

Return type:

list[float]

redshift_values: ndarray[tuple[Any, ...], dtype[float64]]
use_bh_mass: bool = False
use_selection_effects_correction: bool = True
master_thesis_code.bayesian_inference.bayesian_inference.dist_array(redshifts, h=H, Omega_m=OMEGA_M, Omega_de=OMEGA_DE)[source]

Vectorized luminosity distance in Gpc over an array of redshifts.

Delegates to physical_relations.dist_vectorized for a canonical, unit-consistent implementation. Returns Gpc (same unit as the scalar dist()).

Parameters:
Return type:

ndarray[tuple[Any, …], dtype[float64]]