Statistical framework

Statistical framework#

See also

A full description of the statistical framework of CHIMERA is presented in Borghi et al. 2023.

CHIMERA is based on an extension of the Hierarchical Bayesian inference framework presented in Mandel et al. 2019 and Vitale et al. 2022. Consider a population of GW sources, individually described by source-frame parameters \(\boldsymbol{\theta}\) which globally follow a probability distribution described by hyper-parameters \(\boldsymbol{\lambda}\), namely:

  • \(\boldsymbol{\lambda}=\{\boldsymbol{\lambda}_\mathrm{c},\boldsymbol{\lambda}_\mathrm{m},\boldsymbol{\lambda}_\mathrm{z}\}\) population hyper-parameters describing cosmology \((\boldsymbol{\lambda}_\mathrm{c})\), binary mass distribution \((\boldsymbol{\lambda}_\mathrm{m})\) and rate evolution \((\boldsymbol{\lambda}_\mathrm{z})\);

  • \(\boldsymbol{\theta}=\{z,m_1,m_2,\hat{\Omega},\dots\}\) GW event source-frame parameters, including redshift \((z)\), primary mass \((m_1)\), secondary mass \((m_2)\), and sky location \((\hat{\Omega})\);

  • \(\boldsymbol{\theta}^\mathrm{det}=\{d_L,m_1^\mathrm{det},m_2^\mathrm{det},\dots\}\) GW event detector-frame parameters, including luminosity distance \((d_L)\), detected primary mass \((m_1)\), detected secondary mass \((m_2)\).

Given a set \(\boldsymbol{d}^{\rm GW}=\{\boldsymbol{d}^{\rm GW}_i\}~~\left(i=1,\dots,N_{\rm ev}\right)\) of data from independent GW events, the goal of CHIMERA is to evaluate the hyper-likelihood

\[p(\boldsymbol{d}^{\rm GW} | \boldsymbol{\lambda}) \propto \frac{1}{\xi(\boldsymbol{\lambda})^{N_{\rm ev}}} \prod_{i=1}^{N_{\rm ev}} \int \mathrm{d}z\, \mathrm{d}\hat{\Omega} \, \mathcal{K}_{\mathrm{gw},i}(z, \hat{\Omega} | \boldsymbol{\lambda}_\mathrm{c}, \boldsymbol{\lambda}_\mathrm{m}) \, p_{\rm gal}(z, \hat{\Omega} | \boldsymbol{\lambda}_{\rm c})\, \frac{\psi(z ; \boldsymbol{\lambda}_{\rm z})}{1+z}\,, \label{eq:like_full}\]

where the GW kernel \(\mathcal{K}_{\mathrm{gw},i} (z, \hat{\Omega} | \boldsymbol{\lambda}_\mathrm{c}, \boldsymbol{\lambda}_\mathrm{m})\) and the selection bias term \(\xi(\boldsymbol{\lambda})\) are defined as:

\[\begin{split}\begin{align} \mathcal{K}_{\mathrm{gw},i} (z, \hat{\Omega} | \boldsymbol{\lambda}_\mathrm{c}, \boldsymbol{\lambda}_\mathrm{m}) &\equiv \int \mathrm{d}m_1 \mathrm{d}m_2 \, \frac{ p(z, m_1, m_2, \hat{\Omega} | \boldsymbol{d}^{\rm GW}_i)}{ \pi( d_L ) \pi( m_1^\mathrm{det} ) \pi( m_2^\mathrm{det} ) } \, \frac{p(m_1, m_2 | \boldsymbol{\lambda}_{\rm m})}{\frac{\mathrm{d} d_L}{\mathrm{d} z}(z, \boldsymbol{\lambda}_\mathrm{c}) (1+z)^2}\,,\label{eq:Kgw} \\ \xi(\boldsymbol{\lambda}) &\equiv \int \mathrm{d} \boldsymbol{\theta}^\mathrm{det} \, P_{\rm det}(\boldsymbol{\theta}^\mathrm{det})\, \, \frac{p(m_1, m_2 | \boldsymbol{\lambda}_\mathrm{m})}{\frac{\mathrm{d} d_L}{\mathrm{d} z}(z, \boldsymbol{\lambda}_\mathrm{c}) (1+z)^2} \, p_{\rm gal}(z, \hat{\Omega} | \boldsymbol{\lambda}_{\rm c})\, \frac{\psi(z ; \boldsymbol{\lambda}_{\rm z})}{1+z} \,, \label{eq:selection_effects_xi} \end{align}\end{split}\]

[TBD]

In CHIMERA, the GW kernel is computed via kernel density estimation (KDE) method, while the selection bias term is computed via Monte Carlo integration.