Sidestepping the inversion of the weak-lensing covariance matrix with Approximate Bayesian Computation

TL;DR

This paper introduces an Approximate Bayesian Computation method to perform cosmological parameter inference from weak lensing data without inverting singular covariance matrices, enabling analysis in high-dimensional, simulation-limited scenarios.

Contribution

It presents a likelihood-free inference approach using ABC to handle singular covariance matrices in cosmology, especially for large data vectors where traditional methods fail.

Findings

ABC provides unbiased parameter estimates even when covariance matrices are singular.

Parameter estimate variances are mildly larger with ABC compared to likelihood-based methods.

The method is demonstrated on realistic Euclid-like weak lensing data scenarios.

Abstract

Weak gravitational lensing is one of the few direct methods to map the dark-matter distribution on large scales in the Universe, and to estimate cosmological parameters. We study a Bayesian inference problem where the data covariance $\mathbf{C}$, estimated from a number $n_{\textrm{s}}$ of numerical simulations, is singular. In a cosmological context of large-scale structure observations, the creation of a large number of such $N$-body simulations is often prohibitively expensive. Inference based on a likelihood function often includes a precision matrix, $\Psi = \mathbf{C}^{-1}$. The covariance matrix corresponding to a $p$-dimensional data vector is singular for $p \ge n_{\textrm{s}}$, in which case the precision matrix is unavailable. We propose the likelihood-free inference method Approximate Bayesian Computation (ABC) as a solution that circumvents the inversion of the singular covariance matrix. We present examples of increasing degree of complexity, culminating in a realistic cosmological scenario of the determination of the weak-gravitational lensing power spectrum for the upcoming European Space Agency satellite Euclid. While we found the ABC parameter estimate variances to be mildly larger compared to likelihood-based approaches, which are restricted to settings with $p < n_{\textrm{s}}$, we obtain unbiased parameter estimates with ABC even in extreme cases where $p / n_{\textrm{s}} \gg 1$. The code has been made publicly available to ensure the reproducibility of the results.