Leveraging rapid parameter estimates for efficient gravitational-wave Bayesian inference via posterior repartitioning

Technical Summary: Leveraging Rapid Parameter Estimates for Efficient Gravitational-Wave Bayesian Inference via Posterior Repartitioning

Problem Statement
Gravitational-wave (GW) astronomy relies on rigorous Bayesian parameter estimation (PE) to infer physical source properties from observed signals. The standard approach utilizes nested sampling (NS) algorithms, which are robust and provide Bayesian evidence but are computationally expensive, often requiring millions of likelihood evaluations and hours to days of runtime on high-performance computing clusters. As detector sensitivity improves and event rates increase, this computational bottleneck threatens the feasibility of routine analysis, particularly for high signal-to-noise ratio (SNR) events where the posterior volume becomes increasingly small relative to the prior, necessitating more iterations for convergence. While rapid inference methods exist, they are typically not integrated into final high-fidelity analyses because doing so risks violating the statistical requirement that the Bayesian prior remain independent of the data.

Methodology
The authors propose a novel framework that integrates rapid, physically-motivated constraints from the simple-pe algorithm with the nested sampling acceleration technique of posterior repartitioning (PR). The method operates through the following steps:

1. Rapid Initial Constraints: The simple-pe algorithm is run on the GW strain data. Unlike stochastic samplers, simple-pe uses physical decompositions of GW signals (dominant term, spin-induced precession, and higher-order multipoles) to generate initial parameter estimates in minutes on a single CPU.
2. Continuous Distribution Learning: The discrete samples from simple-pe are converted into a continuous probability distribution using a normalizing flow (trained via modified margarine software).
3. Robust Widening: To prevent bias, the learned distribution is "widened" by increasing the standard deviation of its base multivariate Gaussian distribution. This ensures the repartitioned prior covers the true posterior, even if the initial simple-pe estimate is imperfect or misses secondary modes.
4. Posterior Repartitioning: The widened distribution serves as a new, effective prior ($\pi'(\theta)$) for the nested sampler. To maintain statistical rigor, the likelihood is redefined with a corrective term: $L'(\theta) = L(\theta) [\pi(\theta) / \pi'(\theta)]$. This ensures the product of the likelihood and prior remains invariant, guaranteeing that the final posterior is mathematically identical to a standard, uninformed analysis.
5. Execution: The final PE is performed using a custom bilby plugin (bilby-pr) that implements the repartitioned prior and likelihood, utilizing the dynesty sampler.

Key Contributions

• Statistically Rigorous Acceleration: The paper presents a method that leverages low-latency information to accelerate the final inference without compromising the independence of the Bayesian prior, a requirement often violated by other "informed" approaches.
• Integration of Physical Heuristics: By using simple-pe (which relies on physical arguments rather than stochastic search) to initialize the PR, the method avoids the pitfalls of low-resolution preliminary NS runs, which can fail to identify complex posterior structures like secondary modes.
• SNR-Dependent Calibration: The authors empirically calibrate "widening factors" for the repartitioned prior based on network SNR, establishing distinct regimes for low-SNR ($<20$) and high-SNR ($\ge20$) signals to balance efficiency and robustness.

Results
The method was validated through two primary studies:

1. SNR Scaling Study: Using a fiducial binary black hole merger, the authors demonstrated that the computational speedup scales with SNR.
   • For SNR $< 20$, the method can be slower than standard NS due to the broadness of the posterior and the overhead of the repartitioning calculation.
   • For SNR $> 20$, the method becomes increasingly efficient. At SNR 150, the number of likelihood evaluations and total sampling time were reduced by factors of 1.7 and 1.7 respectively (a 41% runtime reduction).
   • At SNR 150, the per-sample speedup (normalized by effective samples) reached 2.1x (a 53% runtime reduction).
   • The number of likelihood evaluations for the PR analysis remained approximately constant for SNR $> 20$, whereas standard NS evaluations grew linearly with SNR.
2. Large-Scale Injection Study: A study of 100 simulated binary black hole mergers confirmed the statistical integrity of the method.
   • Probability-Probability (P-P) plots showed that the method produces unbiased posteriors consistent with standard NS for the majority of signals.
   • The method successfully handled bimodal posteriors (e.g., inclination angle) where simple-pe correctly identified both modes, preventing the repartitioned prior from excluding significant probability regions.
   • Failures were observed primarily at low SNR ($<10$) or in rare cases where simple-pe significantly misidentified the sky mode, leading to offsets that the fixed widening procedure could not fully cover.

Significance and Claims
The paper claims that this method provides a powerful technique to mitigate the computational cost of analyzing signals from current and future GW observatories.

• Scalability: The performance gain scales with SNR, making it particularly valuable for the "loud" events that will become more common with improved detector sensitivity (e.g., in O4, O5, and future 3G detectors or LISA).
• Workflow Integration: The method allows the valuable information from low-latency alerts (like simple-pe) to inform the final, high-fidelity analysis without violating Bayesian principles.
• Robustness: By design, the method sacrifices efficiency when necessary (via conservative widening) to guarantee statistically unbiased inference.
• Future Outlook: While currently recommended for signals with network SNR $> 20$ (which constitute a minority of current detections but a growing fraction of future ones), the authors suggest that automating the widening procedure and extending the method to precessing systems could further improve performance. The method is presented as a complementary technique that can be combined with likelihood acceleration and other sampling optimizations.