Cosmic Inference: Constraining Parameters With Observations and Highly
  Limited Number of Simulations
  Cosmological probes pose an inverse problem where the measurement result is
obtained through observations, and the objective is to infer values of model
parameters which characterize the underlying physical system -- our Universe.
Modern cosmological probes increasingly rely on measurements of the small-scale
structure, and the only way to accurately model physical behavior on those
scales, roughly 65 Mpc/h or smaller, is via expensive numerical simulations. In
this paper, we provide a detailed description of a novel statistical framework
for obtaining accurate parameter constraints by combining observations with a
very limited number of cosmological simulations. The proposed framework
utilizes multi-output Gaussian process emulators that are adaptively
constructed using Bayesian optimization methods. We compare several approaches
for constructing multi-output emulators that enable us to take possible
inter-output correlations into account while maintaining the efficiency needed
for inference. Using Lyman alpha forest flux power spectrum, we demonstrate
that our adaptive approach requires considerably fewer --- by a factor of a few
in Lyman alpha P(k) case considered here --- simulations compared to the
emulation based on Latin hypercube sampling, and that the method is more robust
in reconstructing parameters and their Bayesian credible intervals.
