Papers › Sidestepping the inversion of the weak-lensing covariance matrix with Approximate...
Sidestepping the inversion of the weak-lensing covariance matrix with Approximate Bayesian Computation
Martin Kilbinger, Emille E. O. Ishida, Jessi Cisewski-Kehe
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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 𝐂, estimated from a number nₛ 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, Ψ= 𝐂⁻¹. The covariance matrix corresponding to a p-dimensional data vector is singular for p ≥nₛ, 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 < nₛ, we obtain unbiased parameter estimates with ABC even in extreme cases where p / nₛ ≫1. The code has been made publicly available to ensure the reproducibility of the results.
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