Bayesian control variates for optimal covariance estimation with pairs of simulations and surrogates

Predictions of the mean and covariance matrix summary statistics are critical for confronting cosmological theories with observations, not least likelihood approximations parameter inference. The price to pay accurate estimates is extreme cost running $N$-body hydrodynamics simulations. Approximate solvers, or surrogates, greatly reduce computational but can introduce significant biases, example in non-linear regime cosmic structure growth. We propose "CARPool Bayes", an approach solve inference problem both means covariances using a combination simulations surrogates. Our framework allows incorporating prior information covariance. derive closed-form solutions Maximum A Posteriori that efficient Bayesian shrinkage estimators, guarantee positive semi-definiteness, optionally leverage analytical approximations. discuss choices simple procedure obtaining optimal hyperparameter values small set test our method by estimating clustering GADGET-III at redshift $z=0.5$ surrogates from 100-1000$\times$ faster particle-mesh code. Taking sample 15,000 as truth, empirical Bayes diagonal blocks, estimator produces nearly identical Fisher contours $\Lambda$CDM parameters only $15$ dark matter power spectrum. In this case number so would be degenerate. show cases where even na\"ive still improves estimate. applicable wide range astrophysical problems fast available.