Nuisance hardened data compression for fast likelihood-free inference

We show how nuisance parameter marginalized posteriors can be inferred directly from simulations in a likelihood-free setting, without having to jointly infer the higher dimensional interesting and nuisance parameter posterior first and marginalize a posteriori. The result is that for an inference task with a given number of interesting parameters, the number of simulations required to perform likelihood-free inference can be kept (roughly) the same irrespective of the number of additional nuisances to be marginalized over. To achieve this, we introduce two extensions to the standard likelihood-free inference set-up. First, we show how nuisance parameters can be recast as latent variables and hence automatically marginalized over in the likelihood-free framework. Secondly, we derive an asymptotically optimal compression from N data to n summaries – one per interesting parameter - such that the Fisher information is (asymptotically) preserved, but the summaries are insensitive to the nuisance parameters. This means that the nuisance marginalized inference task involves learning n interesting parameters from n ‘nuisance hardened’ data summaries, regardless of the presence or number of additional nuisance parameters to be marginalized over. We validate our approach on two examples from cosmology: supernovae and weak-lensing data analyses with nuisance parametrized systematics. For the supernova problem, high-fidelity posterior inference of Ωm and w0 (marginalized over systematics) can be obtained from just a few hundred data simulations. For the weak-lensing problem, six cosmological parameters can be inferred from just $\mathcal {O}(10^3)$ simulations, irrespective of whether 10 additional nuisance parameters are included in the problem or not.