Extremely expensive likelihoods: A variational-Bayes solution for
  precision cosmology
  We present a variational-Bayes solution to compute non-Gaussian posteriors
from extremely expensive likelihoods. Our approach is an alternative for
parameter inference when MCMC sampling is numerically prohibitive or
conceptually unfeasible. For example, when either the likelihood or the
theoretical model cannot be evaluated at arbitrary parameter values, but only
previously selected values, then traditional MCMC sampling is impossible,
whereas our variational-Bayes solution still succeeds in estimating the full
posterior. In cosmology, this occurs e.g. when the parametric model is based on
costly simulations that were run for previously selected input parameters. We
demonstrate our posterior construction on the KiDS-450 weak lensing analysis,
where we reconstruct the original KiDS MCMC posterior at 0.6% of its former
numerical cost. The reduction in numerical cost implies that systematic effects
which formerly exhausted the numerical budget could now be included.
