What does a cosmological experiment really measure? Covariant posterior
  decomposition with normalizing flows
  We present methods to rigorously extract parameter combinations that are
constrained by data from posterior distributions. The standard approach uses
linear methods that apply to Gaussian distributions. We show the limitations of
the linear methods for current surveys, and develop non-linear methods that can
be used with non-Gaussian distributions, and are independent of the parameter
basis. These are made possible by the use of machine-learning models,
normalizing flows, to learn posterior distributions from their samples. These
models allow us to obtain the local covariance of the posterior at all
positions in parameter space and use its inverse, the Fisher matrix, as a local
metric over parameter space. The posterior distribution can then be
non-linearly decomposed into the leading constrained parameter combinations via
parallel transport in the metric space. We test our methods on two
non-Gaussian, benchmark examples, and then apply them to the parameter
posteriors of the Dark Energy Survey and Planck CMB lensing. We illustrate how
our method automatically learns the survey-specific, best constrained effective
amplitude parameter $S_8$ for cosmic shear alone, cosmic shear and galaxy
clustering, and CMB lensing. We also identify constrained parameter
combinations in the full parameter space, and as an application we estimate the
Hubble constant, $H_0$, from large-structure data alone.
