Reconstructing and resampling: a guide to utilising posterior samples from gravitational wave observations

Technical Summary: Reconstructing and Resampling: A Guide to Utilising Posterior Samples from Gravitational Wave Observations

Problem Statement
The LIGO–Virgo–KAGRA (LVK) collaboration releases gravitational-wave (GW) data alongside Bayesian posterior distributions for observed signals, typically provided as equally-weighted samples. These samples are essential for downstream analyses, such as constraining the population of merging black holes or testing fundamental physics. However, many such analyses require modifying the underlying assumptions of the original inference—such as changing the waveform model, the prior distribution, or the noise power spectral density (PSD). Re-running the full stochastic sampling process (e.g., Markov Chain Monte Carlo or nested sampling) for every variation is computationally prohibitive. While resampling techniques exist to generate new posterior sets from existing samples without re-analysis, their application in GW astronomy often lacks systematic documentation regarding the accurate reconstruction of the likelihood and prior, and the comparative efficiency of different resampling strategies.

Methodology
The paper outlines a comprehensive framework for reconstructing the posterior density from LVK data products (specifically those generated by the Bilby inference library and packaged via PESummary) and applying resampling techniques.

1. Reconstruction of Likelihood and Prior:

   • The authors detail the requirements to accurately reconstruct the log-likelihood and log-prior from stored posterior samples. This involves matching the original computing environment (specifically conda environments and software versions for Bilby, LALSimulation, etc.) and hardware configurations.
   • Key components for reconstruction include: using the original strain data (preferably at 16384 Hz before downsampling), applying the correct windowing (Tukey window), utilizing the specific PSD estimation method (BayesWave), and configuring the waveform approximant (e.g., IMRPhenomXPHM).
   • The paper distinguishes between marginalised likelihoods (used during sampling) and non-marginalised likelihoods (corresponding to the full posterior), noting that stored values in data releases are typically marginalised.
   • Calibration uncertainties are handled by marginalising over a spline approximation constructed from calibration envelopes.

2. Resampling Techniques:

   • Rejection Sampling (RS): Samples are accepted with a probability proportional to the ratio of the target density to the proposal density. The efficiency depends on the bound $M$ of this ratio.
   • Importance Sampling (IS): Uses weights directly to estimate statistics or create weighted histograms. The paper distinguishes between standard IS (weighted estimates) and multinomial-IS (resampling with replacement to create a new set of samples).
   • Pareto-Smoothing: To address inefficiencies caused by heavy-tailed weight distributions, the authors introduce Pareto-smoothed IS (PSIS) and Pareto-smoothed RS (PSRS). This involves fitting a Generalised Pareto distribution to the largest weights and replacing them with expected order statistics. A diagnostic parameter $\hat{k}$ is used to assess reliability ($\hat{k} < 0.5$ is reliable; $\hat{k} > 0.7$ indicates high bias).

3. Efficiency Metrics:

   • The paper defines efficiency for RS as the acceptance rate and for IS as the Effective Sample Size (ESS) relative to the original sample size.
   • Analytical approximations for efficiency are provided assuming log-normally distributed weights, linking efficiency to the variance of the log-weights.

Key Contributions

• Reconstruction Guide: A practical guide to reconstructing likelihoods and priors from LVK public data, demonstrating that with matched software environments, residuals between reconstructed and stored values are negligible (typically $< 10^{-3}$ for likelihoods), ensuring resampling validity.
• Comparative Analysis: A systematic comparison of RS, multinomial-IS, and their Pareto-smoothed variants. The authors demonstrate that while RS produces independent samples, it can be inefficient if distributions differ significantly. Multinomial-IS offers smoother histograms but introduces sample correlations. Pareto-smoothing improves efficiency for both methods when weight distributions have heavy tails.
• Bayes Factor Estimation: A derivation showing how to estimate the Bayes factor (and its uncertainty) between the original and modified models using the resampling weights, allowing for model comparison without re-analysis.
• Validation: The methods are validated using toy models and real GW events (GW150914 and GW230627_015337).

Results

• Reconstruction Accuracy: Tests on GW150914 and GW230627_015337 show that reconstructed log-likelihoods and log-priors match stored values with high precision. Hardware differences (e.g., Intel Xeon vs. Apple M3) introduce random shifts of $\approx 3 \times 10^{-6}$, which are negligible for resampling.
• Resampling Performance:
  • In toy models where the proposal distribution is broader than the target (under-constrained), both RS and IS perform well.
  • When the proposal is narrower (over-constrained), RS efficiency drops rapidly. IS degrades more gradually but suffers from high variance unless smoothed.
  • Pareto-smoothing significantly improves efficiency in over-constrained scenarios, provided the diagnostic $\hat{k}$ remains below 0.7.
• Application Examples:
  • Waveform Model Change: Resampling GW150914 from the IMRPhenomXPHM model to the IMRPhenomXP model (removing higher-order modes) successfully reproduced the posterior of a full re-analysis. The resulting Bayes factor ($\ln B = 0.16 \pm 0.03$) indicated mild support for the simpler model.
  • PSD Update: Resampling with an upgraded BayesWave PSD (GWTC-4.0 settings) showed a few-percent shift in the chirp mass posterior. The analysis of Bayes factors revealed that while the new PSD assigned lower overall probability to the data, it provided a noise model that made the signal more distinguishable.

Significance and Claims
The paper positions itself as a practical resource for the GW community, aiming to lower the barrier for researchers to perform "what-if" analyses without access to high-performance computing resources required for full re-analysis. The authors claim that:

1. Accurate reconstruction of posteriors is achievable across different computing environments, provided software versions are matched.
2. Resampling is a reliable alternative to re-analysis for modest changes in assumptions (e.g., waveform updates, PSD refinements, prior adjustments).
3. The introduction of Pareto-smoothing diagnostics provides a necessary tool to assess the validity of resampling results, preventing the use of biased estimates in extreme cases.
4. The guide enables systematic uncertainty assessments and population studies that leverage the full wealth of LVK data releases, extending the scientific utility of existing posterior samples.

The authors remain modest regarding the scope, noting that the method relies on Bilby-generated posteriors and that extreme modifications (e.g., changing waveform families entirely or drastically altering frequency bands) may still necessitate full re-analysis. They emphasize that while resampling is powerful, it is not a universal substitute for re-analysis when the proposal and target distributions diverge too significantly.