Sparse Bayesian mass-mapping with uncertainties

Abstract

Mass-mapping via weak gravitational lensing has until recently lacked principled statistical consideration of uncertainties introduced during the reconstruction process – solving of an often seriously ill-posed inverse problem. In recent work we posed the mass-mapping inverse problem as an unconstrained Bayesian inference problem with Laplace-type $\ell_1$-norm sparsity-promoting prior, which we solve via convex optimisation. Formulating the problem in this way allows us to exploit recent developments in probability concentration theory to infer tightly bound, theoretically conservative uncertainties $\mathcal{O}(10^6)$ times faster than traditional MCMC techniques. Building on these new fast Bayesian inference techniques we have developed several uncertainty quantification techniques primarily aimed towards the gravitational lensing paradigm, though entirely generalizable to other settings. The uncertainty quantification techniques reviewed here are; knock-out hypothesis testing of structure, local credible regions (cf. pixel-level Bayesian error bars), and Bayesian locational uncertainty of structure. Additionally, these conservative Bayesian inferences can be leveraged to aggregate uncertainties which are often computed by the weak lensing community (e.g. peak statistics).