  In this brief paper we revisit the Fisher information content of cosmological
power spectra or two-point functions of Gaussian fields in order to comment on
the assumption of Gaussian estimators and the use of parameter-dependent
covariance matrices for parameter inference in the context of precision
cosmology. Even though the assumption of a Gaussian likelihood is motivated by
the central limit theorem, we discuss that it leads to Fisher information
content that violates the Cram\'er-Rao bound if used consistently, owing to
independent but artificial information from the parameter-dependent covariance
matrix. At any fixed multipole, this artificial term is shown to become
dominant in the case of a large number of correlated fields. While the
distribution of the estimators does indeed tend to a Gaussian with a large
number of modes, it is shown, however, that its Fisher information content does
not, in the sense that their covariance matrix never carries independent
information content, precisely because of the non-Gaussian shape of the
distribution. In this light, we discuss the use of parameter-dependent
covariance matrices with Gaussian likelihoods for parameter inference from
two-point statistics. As a rule of thumb, Gaussian likelihoods should always be
used with a covariance matrix fixed in parameter space, since only this
guarantees that conservative information content is assigned to the
observables, and at the same time prevents biases appearing.
