  The recovery of an exoplanet's atmospheric parameters from its spectrum
requires accurate knowledge of the spectral errors and covariances.
Unfortunately, the complex image processing used in high-contrast
integral-field spectrograph (IFS) observations generally produces spectral
covariances that are poorly understood and often ignored. In this work, we show
how to measure the spectral errors and covariances and include them
self-consistently in parameter retrievals. By combining model exoplanet spectra
with a realistic noise model generated from GPI early science data, we show
that ignoring spectral covariance in high-contrast IFS data can both bias
inferred parameters and lead to unreliable confidence regions on those
parameters. This problem is made worse by the common practice of scaling the
$\chi^2$ per degree of freedom to unity; the input parameters then fall outside
the $95\%$ confidence regions in as many as ${\sim}80\%$ of noise realizations.
The biases we observe can approach the typical levels of precision achieved in
high-contrast spectroscopy. Accounting for realistic priors in fully Bayesian
retrievals can also have a significant impact on the inferred parameters.
Plausible priors on effective temperature and surface gravity can vary by an
order of magnitude across the confidence regions appropriate for objects with
weak age constraints; priors for objects with good age constraints are
dominated by modeling uncertainties. Our methods are directly applicable to
existing high-contrast IFSs including GPI and SPHERE, as well as upcoming
instruments like CHARIS and, ultimately, WFIRST-AFTA.
