The Parameter-Level Performance of Covariance Matrix Conditioning in
  Cosmic Microwave Background Data Analyses
  Empirical estimates of the band power covariance matrix are commonly used in
cosmic microwave background (CMB) power spectrum analyses. While this approach
easily captures correlations in the data, noise in the resulting covariance
estimate can systematically bias the parameter fitting. Conditioning the
estimated covariance matrix, by applying prior information on the shape of the
eigenvectors, can reduce these biases and ensure the recovery of robust
parameter constraints. In this work, we use simulations to benchmark the
performance of four different conditioning schemes, motivated by contemporary
CMB analyses. The simulated surveys measure the $TT$, $TE$, and $EE$ power
spectra over the angular multipole range $300 \le \ell \le 3500$ in $\Delta
\ell = 50$ wide bins, for temperature map-noise levels of $10, 6.4$ and
$2\,\mu$K-arcmin. We divide the survey data into $N_{\mathrm{real}} = 30, 50,$
or 100 uniform subsets. We show the results of different conditioning schemes
on the errors in the covariance estimate, and how these uncertainties on the
covariance matrix propagate to the best-fit parameters and parameter
uncertainties. The most significant effect we find is an additional scatter in
the best-fit point, beyond what is expected from the data likelihood. For a
minimal conditioning strategy, $N_{\mathrm{real}} = 30$, and a temperature
map-noise level of 10$\,\mu$K-arcmin, we find the uncertainty on the recovered
best fit parameter to be $\times 1.3$ larger than the apparent posterior width
from the likelihood ($\times 1.2$ larger than the uncertainty when the true
covariance is used). Stronger priors on the covariance matrix reduce the
misestimation of parameter uncertainties to $<1\%$. As expected, empirical
estimates perform better with higher $N_{\mathrm{real}}$, ameliorating the
adverse effects on parameter constraints.
