Gravitational Wave Localization Posteriors

• Gravitational wave localization posteriors are Bayesian probability distributions that describe the sky and 3D positions of GW sources using detector data and astrophysical priors.
• The SKYFAST pipeline employs a Dirichlet Process Gaussian Mixture Model to reconstruct analytic posteriors in real time, achieving convergence in 20–36% of the total runtime.
• These advanced posteriors facilitate rapid electromagnetic follow-up, accurate host galaxy ranking, and improved cosmological inferences through efficient marginalization and conditioning of parameters.

A gravitational wave localization posterior is the Bayesian probability distribution describing the sky position—and, increasingly, the three-dimensional location—of a gravitational-wave (GW) source, conditioned on data from a detector network and an astrophysical model. Such posteriors are foundational for the design and success of electromagnetic follow-up, for host identification, and for cosmological inference using GW sources as standard sirens. They integrate detector response, parameter estimation uncertainties, and astrophysical priors, and their construction spans numerical, analytical, and non-parametric methodologies.

1. Construction of Localization Posteriors and the SKYFAST Pipeline

Gravitational-wave localization posteriors are typically generated from samples obtained via full parameter estimation (PE) pipelines—such as Bilby-MCMC or nested sampling—wherein each sample contains values for sky location (right ascension α, declination δ), luminosity distance (dₗ), and inclination angle (θ_jn), among other parameters. The SKYFAST pipeline is designed to operate in parallel with such PE analyses. It ingests the evolving stream of posterior samples to construct an analytic, up-to-date model of the multidimensional localization posterior, leveraging only a fraction of the eventual full sample ensemble.

SKYFAST implements its posterior reconstruction via a Dirichlet Process Gaussian Mixture Model (DPGMM), as instantiated in the FIGARO code. This Bayesian non-parametric approach produces a mixture model for the sampled parameters, updating the analytical posterior over (α, δ, dₗ, θ_jn) in real time as new samples become available. SKYFAST includes logic to monitor convergence using the evolution of the information entropy S(N) of the reconstructed density. Once S(N) stabilizes (as judged by three zero crossings of its derivative), the reconstructed analytic posterior is deemed stable, and an “intermediate” localization can be published.

This approach allows the analytical localization to be released significantly before the full PE sampling concludes; typically, 20–36% of the total runtime is sufficient for accurate localization, according to the convergence criterion based on information entropy (Demasi et al., 18 Jul 2024).

2. Analytical Posterior Reconstruction Using the Dirichlet Process Gaussian Mixture Model

The core of SKYFAST’s localization posterior is the DPGMM. In the DPGMM formalism, the target density over input space x = (α, δ, dₗ, θ_jn) is approximated as:

$$p(x) \approx \sum_{k=1}^{N_G} w_k \mathcal{N}(x \mid \mu_k, \Sigma_k)$$

where $N_G$ is the (inferred) number of mixture components, $w_k$ the mixing weights, $\mu_k$ the component means, and $\Sigma_k$ the component covariance matrices. The Dirichlet process prior admits variable $N_G$ and enables the model to adapt to complex, multimodal, anisotropic, or degenerate structures in the PE sample stream.

As new samples arrive, the DPGMM updates the mixture parameters, and the full joint posterior can be analytically marginalized or conditioned as required. The non-parametric nature of the DPGMM allows for maximal flexibility in capturing posterior features such as angular–distance degeneracy, multimodality, or the specific conditioning of inclination angle on host distance.

The information entropy $S(N) = - \sum_k p_k^{(N)} \log p_k^{(N)}$ is evaluated on a discrete grid of the reconstructed posterior, and convergence is claimed when $dS/dN$ exhibits several (typically three) zero crossings. Once converged, the analytic mixture captures the essential structure of the posterior, supporting accurate quantification of localization regions and derived probabilities (Demasi et al., 18 Jul 2024).

3. Time-Efficient Localization and Convergence Monitoring

A central innovation of SKYFAST is its ability to provide intermediate localization products well before the PE sampling run completes. Because the DPGMM requires only a fraction of the total PE samples to accurately reconstruct the main posterior features, actionable skymaps and host galaxy rankings can be released after 20–36% of the total run time, depending on the sampler and problem complexity (Demasi et al., 18 Jul 2024). The emission of these products is regulated by the entropy-based convergence metric; an “intermediate” posterior may be released rapidly, and subsequently refined as more samples become available, until the final PE result is achieved.

This reduction in latency is critical for electromagnetic (EM) follow-up campaigns: rapid skymap release significantly increases the probability of capturing prompt transient emission, particularly in the context of short–gamma-ray burst afterglows or kilonovae. The entropy criterion ensures that posteriors are not released prematurely or in a state of unconverged structural bias.

4. Host Galaxy Ranking and Conditional Inclination Posterior

Once the analytic posterior is built, SKYFAST efficiently evaluates the marginalized three-dimensional probability density at the positions $(α_g, δ_g, d_{l, g})$ of galaxies drawn from an external catalog (such as GLADE+). This enables the generation of a ranked list of probable host galaxies, restricted to those within, e.g., the 90% credible volume.

A key feature is SKYFAST’s ability to report, for each candidate host, the posterior distribution of the inclination angle θ_jn, conditioned explicitly on the galaxy’s known position and distance. This conditional posterior is sharper than the marginal distribution, as it exploits the breaking of the natural $d_l$–$θ_jn$ degeneracy provided by the galaxy’s known $d_l$. This refined inclination posterior is particularly useful for assessing gamma-ray burst detectability, where jet orientation strongly modulates observed emission properties.

5. Implications for Multi-Messenger Astronomy and Cosmology

By providing rapid, robust localization posteriors—including volume uncertainty and orientation constraints—SKYFAST enhances the efficiency of multi-messenger astronomy. Early, accurate host rankings allow for optimal allocation of telescope resources, minimizing the time to identify electromagnetic counterparts and supporting rapid target-of-opportunity (ToO) observations.

Furthermore, the analytical posterior enables cross-correlation with large-scale structure and galaxy redshift catalogs, facilitating standard siren measurements of the Hubble constant and other cosmological parameters. Enhanced accuracy and speed in localization directly improve the scientific return of gravitational-wave observations.

6. Comparison to Preceding Frameworks and Outlook

SKYFAST extends the DPGMM-based localization approach previously developed in FIGARO (Rinaldi et al., 2022), fusing its live, sample-driven updates and information entropy–based convergence with PE sample streams. This non-parametric, analytic framework avoids the limitations of pixel-based or histogram-based posteriors (such as those generated by bayestar or LALInference), offering closed-form density evaluation, rapid marginalization/conditioning, and robust finite-sample behavior.

By requiring only a fraction of the total PE samples—without sacrificing accuracy as certified by self-consistency checks (pp-plots) and information entropy stabilization—SKYFAST offers a pragmatic solution for low-latency, high-fidelity gravitational-wave localization. Its extension to future, more efficient Hamiltonian Monte Carlo engines is anticipated as waveform models become fully differentiable.

This methodology exemplifies the evolution toward rapid, robust, and information-rich localization posteriors, positioning gravitational-wave astronomy for maximal scientific impact in upcoming observing runs and in the era of routine multi-messenger observations (Demasi et al., 18 Jul 2024, Rinaldi et al., 2022).