Bayesian optimisation for likelihood-free cosmological inference
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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 propose 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, our strategy combines Gaussian process regression of the discrepancy to build a surrogate surface with Bayesian optimisation to actively acquire training data. We derive and make use of an acquisition function tailored for the purpose of minimising the expected uncertainty in the approximate posterior density. The resulting algorithm (Bayesian optimisation for likelihood-free inference, BOLFI) is applied to the problems of summarising 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.
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