  Parameter inference with an estimated covariance matrix systematically loses
information due to the remaining uncertainty of the covariance matrix. Here, we
quantify this loss of precision and develop a framework to hypothetically
restore it, which allows to judge how far away a given analysis is from the
ideal case of a known covariance matrix. We point out that it is insufficient
to estimate this loss by debiasing a Fisher matrix as previously done, due to a
fundamental inequality that describes how biases arise in non-linear functions.
We therefore develop direct estimators for parameter credibility contours and
the figure of merit. We apply our results to DES Science Verification weak
lensing data, detecting a 10% loss of information that increases their
credibility contours. No significant loss of information is found for KiDS. For
a Euclid-like survey, with about 10 nuisance parameters we find that 2900
simulations are sufficient to limit the systematically lost information to 1%,
with an additional uncertainty of about 2%. Without any nuisance parameters
1900 simulations are sufficient to only lose 1% of information. We also derive
an estimator for the Fisher matrix of the unknown true covariance matrix, two
estimators of its inverse with different physical meanings, and an estimator
for the optimally achievable figure of merit. The formalism here quantifies the
gains to be made by running more simulated datasets, allowing decisions to be
made about numbers of simulations in an informed way.
