On the use of the Derivative Approximation for Likelihoods for Gravitational Wave Inference

TL;DR

This study benchmarks fast, accurate posterior estimation for gravitational-wave inference by comparing the Fisher Matrix, Derivative Approximation for Likelihoods (DALI) in its doublet and triplet forms, and full MCMC, focusing on Einstein Telescope-scale forecasts. Using 300 aligned-spin BBH injections with IMRPhenomHM, the authors show that doublet-DALI delivers substantial speedups with competitive high-dimensional accuracy, while triplet-DALI provides marginal gains for some parameters; a singlet-DALI hybrid offers fast 1D accuracy but limited higher-dimensional fidelity. The GWDALI 1.0 code enables automatic differentiation and modern waveforms, improving stability and performance of higher-order derivatives. Overall, DALI emerges as a practical tool for rapid GW forecast studies and scalable posterior estimation in the ET era and beyond, bridging the gap between simple Fisher forecasts and computationally expensive full MCMC.

Abstract

Posterior inference on the more than a dozen parameters governing a gravitational wave (GW) event is challenging. A typical MCMC analysis can take around $100$ CPU hours, and next generation GW observatories will detect many thousands of events. Here we present a thorough comparison of the accuracy and computational cost of the Fisher Matrix, Derivative Approximation for Likelihoods (DALI) and traditional MCMC methods. We find that using DALI, which extends the traditional Fisher Matrix (FM) method to higher orders, allows for a good approximation of the posterior with a $55$ times smaller computational cost, and that the cost-benefit of the doublet-DALI is better than that of the triplet-DALI. We also show that the singlet-DALI, a hybrid MCMC-Fisher method, is much more accurate than the traditional FM and 10 times faster than the doublet-DALI. A large effort has been invested in forecasting the science case of different detector configurations, and the ability of making fast yet accurate estimations of the posteriors is an important step forward. We also introduce version \texttt{1.0} of the public \texttt{GWDALI} code, which incorporates automatic differentiation, modern waveforms and an optimized parameter decomposition.

Figures (11)

• Figure 1: Correlation matrix $\rho_{ij}$ for the 11 parameters and the 300 GW detections here considered, using the exact posteriors. We show the average $\rho_{ij}$ coefficients (left panel), as well as the 5% lowest and 5% highest (right panel) values for each coefficient. As it is well-known, the pair $d_L$, $\iota$ can be highly correlated or anti-correlated, and the pair $\chi_s$, $\chi_a$ are always highly anti-correlated.
• Figure 2: Two-dimensional $d_L-\iota$ posterior for a particular GW, comparing the exact result (black shaded) with the standard Fisher with Gaussian priors, the MCMC Fisher with exact priors and the DALI approximations.
• Figure 3: Distributions of parameter uncertainties obtained from Fisher. We compare three cases: using the inversion of the raw FM, using the inversion of FM + $F_{\rm prior}$ and through a MCMC sampling of the exact Gaussian likelihood with non-Gaussian priors.
• Figure 4: Comparison of the standard deviation realizations for the marginalized posterior in each parameter. We see that the FM tends to overestimate the uncertainties, while the triplet yields the most accuracy. The doublet using $1/d_L$ is a good compromise in general.
• Figure 5: Median and 68% quantiles 1-D Jenson-Shannon Divergences between the exact posterior and each of the approximations for the marginalized posteriors in each of the individual parameters for the 300 injections. We see that both FM inversion and the doublet DALI approximation using $d_L$ as parameter are the most inaccurate. We also find that the doublet-DALI with $1/d_L$ results in small JSD, comparable to the triplet using $d_L$.