Bayesian optimization for likelihood-free cosmological inference

Many cosmological models have only a finite number of parameters of interest, but a very expensive data-generating process and an intractable likelihood function. We address the problem of performing likelihood-free Bayesian inference from such black-box simulation-based models, under the constraint of a very limited simulation budget (typically a few thousand). To do so, we adopt an approach based on the likelihood of an alternative parametric model. Conventional approaches to approximate Bayesian computation such as likelihood-free rejection sampling are impractical for the considered problem, due to the lack of knowledge about how the parameters affect the discrepancy between observed and simulated data. As a response, we make use of a strategy previously developed in the machine learning literature (Bayesian optimization for likelihood-free inference, BOLFI), which combines Gaussian process regression of the discrepancy to build a surrogate surface with Bayesian optimization to actively acquire training data. We extend the method by deriving an acquisition function tailored for the purpose of minimizing the expected uncertainty in the approximate posterior density, in the parametric approach. The resulting algorithm is applied to the problems of summarizing Gaussian signals and inferring cosmological parameters from the joint lightcurve analysis supernovae data. We show that the number of required simulations is reduced by several orders of magnitude, and that the proposed acquisition function produces more accurate posterior approximations, as compared to common strategies.