  The subject of this paper is a quantification of the information content of
cosmological probes of the large-scale structures, specifically of temperature
and polarisation anisotropies in the cosmic microwave background, CMB-lensing,
weak cosmic shear and galaxy clustering, in terms of Information theory
measures like information entropies. We aim to establish relationships for
Gaussian likelihoods, between conventional measures of statistical
uncertainties and information entropies. Furthermore, we extend these studies
to the computation of (Bayesian) evidences and the power of measurement to
distinguish between competing models. We investigate in detail how cosmological
data decreases information entropy by reducing statistical errors and by
breaking degeneracies. In addition, we work out how tensions between data sets
increase information entropy and quantify this effect in three examples: the
discrepancy in $\Omega_m$ and $\sigma_8$ between the CMB and weak lensing, the
role of intrinsic alignments in weak lensing data when attempting the dark
energy equation of state parameters, and the famous $H_0$-tension between
Cepheids in the Hubble keystone project and the cosmic microwave background as
observed by Planck.
