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

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.