RNLE: Residual neural likelihood estimation and its application to gravitational-wave astronomy

TL;DR

RNLE introduces Residual Neural Likelihood Estimation to gravitational-wave parameter estimation under non-Gaussian noise by learning the residual likelihood from noise realizations and subtracting waveform realizations to evaluate L(d m). It integrates with Bilby via the sbi framework and autoregressive flows to enable simulation-based Bayesian inference on residuals, achieving Gaussian-likelihood consistency in quiet data and superior robustness in glitch-contaminated data. The paper validates RNLE across toy models, simulated BBHs in colored Gaussian and quasi-Gaussian real data, and highly non-Gaussian segments, while diagnosing training-noise variability and proposing ensemble-based strategies. An open-source sbilby implementation provides a practical pathway to deploy RNLE broadly for GW astronomy and other scientific domains requiring flexible, likelihood-free inference under complex noise.

Abstract

Simulation-based inference provides a powerful framework for Bayesian inference when the likelihood is analytically intractable or computationally prohibitive. By leveraging machine-learning techniques and neural density estimators, it enables flexible likelihood or posterior modeling directly from simulations. We introduce Residual Neural Likelihood Estimation (RNLE), a modification of Neural Likelihood Estimation (NLE) that learns the likelihood of non-Gaussian noise in gravitational-wave detector data. Exploiting the additive structure of the signal and noise generation processes, RNLE directly models the noise distribution, substantially reducing the number of simulations required for accurate parameter estimation and improving robustness to realistic noise artifacts. The performance of RNLE is demonstrated using a toy model, simulated gravitational-wave signals, and real detector noise from ground based interferometers. Even in the presence of loud non-Gaussian transients, glitches, we show that RNLE can achieve reliable parameter recovery when trained on appropriately constructed datasets. We further assess the stability of the method by quantifying the variability introduced by retraining the conditional density estimator on statistically identical datasets with different optimization seeds, referred to as training noise. This variability can be mitigated through an ensemble approach that combines multiple RNLE models using evidence-based weighting. An implementation of RNLE is publicly available in the sbilby package, enabling its deployment within gravitational-wave astronomy and a broad range of scientific applications requiring flexible, simulation-based likelihood estimation.

Figures (18)

• Figure 1: Logarithm of the Jensen Shannon divergence value for the posterior distributions obtained with the likelihoods trained with NLE, in green, and RNLE, in orange. The squares stand for the noise related parameter $\sigma$, the circles, triangles, and crosses for the variance, frequency, and amplitude of the sine-Gaussian signal, respectively. The 2-dimensional results are shown in the left panel, while the right panel shows the 4-dimensional results.
• Figure 2: Likelihood evaluation time in milliseconds for NLE, green, and RNLE, orange, versus the number of simulations used in the training. We employ two free parameters in the left panel and four in the right one.
• Figure 3: Schematic overview of the workflow we employ to produce posterior distributions from a set of training data realizations and an observation using RNLE. The pre-processing for the observations, top row, and the training data, bottom row, is the same. We input the training data into the SBI conditional density estimator, which trains a likelihood. This is used in sbilby with the dynesty sampler to perform inference and obtain posterior distributions. At every evaluation of the likelihood, before passing the data to the RNLE likelihood object we subtract a whitened GW signal realization from it.
• Figure 4: Left: Comparison of the posterior distribution of the chirp mass $\mathcal{M}$ obtained using the RNLE likelihood trained using 3000 simulations of coloured-Gaussian noise, in orange, and the Whittle likelihood, in blue. The JS divergence value is reported in the title, confirming the agreement between the two results. Right: Full posterior of the RNLE analysis. The histograms show the posterior distributions for the noise scaling parameter $\sigma$ and the chirp mass. The dash-dotted black line in the histogram shows the true value of the parameter.
• Figure 5: In orange, the logarithm base ten of the JS divergence value for the chirp mass posteriors, computed between the Whittle and RNLE results, against the number of simulations employed to train the conditional density estimator. The blue dotted line indicates the JS divergence threshold signifying the statistical equivalence between the two distributions. The black squares show the RNLE likelihood training time against the number of simulations. We use a logarithmic scale for the $x$-axis and the right $y$-axis. The errorbars are computed over five realizations of re-training and sampling. The green line is obtained using the optimal values from a linear fit in log-log space for the training time versus the number of simulations.