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.
