Bayesian optimization for likelihood-free cosmological inference

Abstract

Many cosmological models have only a finite number of parameters of interest, but a very expensive datagenerating process and an intractable likelihood function. We address the problem of performing likelihoodfree 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 optimisation for likelihood-free inference, bolfi), which combines Gaussian process regression of the discrepancy to build a surrogate surface with Bayesian optimisation to actively acquire training data. We extend the method by deriving an acquisition function tailored for the purpose of minimising the expected uncertainty in the approximate posterior density, in the parametric approach. The resulting algorithm 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.

Parameter inference for weak lensing using Gaussian Processes and MOPED

“…Here we choose to build a GP for each band power, giving 24 GPs. Alternatively, for likelihood-free inference methods, one can also emulate the likelihood directly using the GPs (see Leclercq (2018) and Fendt & Wandelt (2007a)). For power spectrum reconstruction, one can use the PICO method or an alternative, but constrictive, stance is to adopt the approach taken by Habib et al (2007) to first learn a set of basis functions via Singular Value Decomposition (SVD) and model the resulting weights by a Gaussian Process.…”

“…Fendt & Wandelt (2007a) extended the PICO formalism to fit a likelihood function. Leclercq (2018) developed the BOLFI algorithm (Gutmann & Corander 2015) to construct a likelihood regressor which fits two cosmological parameters w 0 and Ω m for the JLA dataset (Betoule et al 2014). In particular, in the BOLFI approach, the uncertainty from the Gaussian Process appears in the acquisition function, which is used to choose the next design point where the simulation will be run.…”

Likelihood Methods for CMB Experiments

“…ABC methods have been initially employed in various fields as a way to bypass the evaluation of the likelihood function with the use of simulated data [72][73][74][75][76][77][78][79][80][81]. Recent applications to astrophysics and cosmology include [82][83][84][85][86]. Similarly to ABC, the simulation-based approach approximates the likelihood by comparing simulated data with observed data using the QML approximations to the ML points of the spectra as a distance metric.…”

Primordial power spectrum and cosmology from black-box galaxy surveys

“…Ishida et al ., 2015;Akeret et al ., 2015;Jennings & Madigan, 2017). More sophisticated approaches known as delfi (Alsing, Wandelt & Feeney, 2018) and bolfi (Leclercq, 2018) have also been recently introduced in cosmology. The ABC approach introduced in this work differs from all of the above in two aspects: (i ) it allows the treatment of a much larger number of parameters (one hundred in this work), which correspond to primordial power spectrum amplitudes at different wavenumbers.…”