Parameter inference and model comparison using theoretical predictions
  from noisy simulations
  When inferring unknown parameters or comparing different models, data must be
compared to underlying theory. Even if a model has no closed-form solution to
derive summary statistics, it is often still possible to simulate mock data in
order to generate theoretical predictions. For realistic simulations of noisy
data, this is identical to drawing realizations of the data from a likelihood
distribution. Though the estimated summary statistic from simulated data
vectors may be unbiased, the estimator has variance which should be accounted
for. We show how to correct the likelihood in the presence of an estimated
summary statistic by marginalizing over the true summary statistic in the
framework of a Bayesian hierarchical model. For Gaussian likelihoods where the
covariance must also be estimated from simulations, we present an alteration to
the Sellentin-Heavens corrected likelihood. We show that excluding the proposed
correction leads to an incorrect estimate of the Bayesian evidence with JLA
data. The correction is highly relevant for cosmological inference that relies
on simulated data for theory (e.g. weak lensing peak statistics and simulated
power spectra) and can reduce the number of simulations required.
