Comparing explicit likelihood and likelihood-free simulation-based inference for weak lensing cosmic shear

Technical Summary: Comparing Explicit Likelihood and Likelihood-Free Simulation-Based Inference for Weak Lensing Cosmic Shear

Problem Statement
Stage-IV weak-lensing (WL) surveys, such as Euclid and LSST, aim to extract precision cosmological parameters (specifically $\Omega_m$ and $S_8$) from non-Gaussian observables. While current analyses predominantly rely on two-point statistics (shear-2PCFs) combined with Gaussian likelihood approximations, this approach compresses the lensing field and fails to fully capture information from non-linear matter evolution. As summary statistics become more informative and non-linear, the Gaussian likelihood assumption becomes increasingly unjustified, potentially leading to biased constraints. Conversely, likelihood-free inference (LFI) offers a way to learn the likelihood directly from simulations using neural density estimators (NDEs), bypassing explicit likelihood modeling. However, a systematic comparison between Explicit Likelihood Inference (ELI) and LFI on identical simulation suites, using matched observables and compression strategies, has been marginally explored. This work addresses the gap in understanding whether discrepancies between frameworks arise from statistical assumptions, numerical implementation, or the information content of the summary statistics.

Methodology
The authors utilize a common suite of Euclid-like simulations generated via the GLASS framework, producing full-sky Gaussian shear maps with shape noise injection representative of the final Euclid Data Release 3 (DR3). The analysis focuses on a non-tomographic configuration with a single redshift bin extending to $z_{max}=3$.

1. Data Generation & Compression:

   • Simulations: Latin Hypercube Sampling (LHS) is used to vary $\Omega_m$ and $\sigma_8$. Two independent sets of 5,100 cosmological nodes are generated: one for deep learning training (LHS0) and one for emulator training and LFI analysis (LHS1).
   • Summary Statistics: Two probes are measured on flat-sky patches:
     • Shear two-point correlation functions ($\xi_\pm$), a Gaussian probe.
     • A map-level Convolutional Neural Network (CNN) statistic using a ResNet-18 architecture, a non-linear probe.
   • Compression: High-dimensional data vectors (DVs) are compressed to the parameter space dimension ($n_p=2$) using two methods:
     • MOPED: A linear, Fisher-optimal compression using numerical derivatives and the covariance matrix.
     • Neural Network (NN) Compression: A non-linear Multilayer Perceptron (MLP) trained to map DVs to parameters.

2. Inference Frameworks:

   • Explicit Likelihood Inference (ELI): Employs Gaussian Process (GP) emulators to predict the mean signal of the compressed statistics. The likelihood is assumed to be multivariate Gaussian, constructed using the GP prediction and an estimated covariance matrix. Posterior sampling is performed via MCMC.
   • Likelihood-Free Inference (LFI): Uses Neural Density Estimators (NDEs) to learn the likelihood or posterior directly from the simulation pairs. The study compares Neural Likelihood Estimation (NLE) and Neural Posterior Estimation (NPE) using architectures such as Neural Spline Flows (NSF), Masked Autoregressive Flows (MAF), and Mixture Density Networks (MDN).

3. Validation:

   • The study employs posterior calibration diagnostics, specifically the Test of Accuracy with Random Points (TARP), to assess the reliability of inferred posteriors.
   • Gaussianity of the compressed summaries is tested using Kolmogorov–Smirnov (KS) tests and global $\chi^2$ goodness-of-fit tests.

Key Results

• Calibration Discrepancies: ELI becomes strongly miscalibrated in the presence of emulation inaccuracies or when the likelihood is non-Gaussian. In contrast, LFI remains consistently well-calibrated across all tested scenarios.
• Impact of Compression: The choice of compression scheme significantly affects ELI performance. While MOPED-compressed shear-2PCFs are close to Gaussian, NN-compressed summaries exhibit strong non-Gaussianity. Consequently, ELI constraints on $\Omega_m$ and $S_8$ can differ by up to a factor of two from those inferred with the CNN statistic due to the failure of the Gaussian likelihood assumption in ELI.
• LFI Robustness: When the issues of emulation error and likelihood non-Gaussianity are addressed, the disagreement between ELI and LFI largely disappears. For LFI, the discrepancy between constraints derived from shear-2PCFs and the CNN statistic is reduced to approximately 30%, highlighting the robustness of the deep-learning probe.
• Gaussianity Tests: The study reveals that while global $\chi^2$ tests may pass for certain compressed summaries, per-bin marginal KS tests can reveal significant non-Gaussianity (e.g., in NN-compressed shear-2PCFs), which is sufficient to invalidate the Gaussian likelihood assumption used in ELI.

Significance and Claims
The authors conclude that in their specific setup—which neglects systematic biases—LFI provides a more robust and better-calibrated framework for extracting cosmological information from weak lensing surveys compared to traditional ELI. The paper emphasizes that accurate non-Gaussian likelihood modeling and rigorous posterior calibration diagnostics are essential for future ELI analyses. The results underscore the potential of deep-learning probes (like CNNs) combined with likelihood-free methods to capture complex information beyond the two-point level without relying on potentially flawed Gaussian approximations. The study serves as a validation framework and offers guidance for the application of ELI and LFI in future Stage-IV WL surveys.