Bias on Tensor-to-Scalar Ratio Inference With Estimated Covariance
  Matrices
  We investigate simulation-based bandpower covariance matrices commonly used
in cosmological parameter inferences such as the estimation of the
tensor-to-scalar ratio~$r$. We find that upper limits on $r$ can be biased low.
The underestimation of the upper limit is most severe when the number of
simulation realizations is similar to the number of observables. Convergence of
the covariance-matrix estimation can require a number of simulations an order
of magnitude larger than the number of observables. This is found to be caused
by an additional scatter in the posterior probability of $r$ due to Monte Carlo
noise in the estimated bandpower covariance matrix, in particular, by spurious
non-zero off-diagonal elements. We show that matrix conditioning can be a
viable mitigation strategy in the case that legitimate covariance assumptions
can be made.
