  Electromagnetic observations have been used over the past decades to
understand the nature of black holes and the material around them. Our ability
to learn about the fundamental physics relies on our understanding of two key
ingredients in the modeling of these electromagnetic observations: the gravity
theory that describes the black hole, and the astrophysics that produces the
observed radiation. In this work we study our current ability to constrain and
detect deviations from General Relativity using the accretion disk spectrum of
stellar-mass black holes in binary systems. Our analysis combines relativistic
ray-tracing and Markov-Chain Monte-Carlo sampling techniques to determine how
well such tests of General Relativity can be carried out in practice. We show
that even when a very simple astrophysical model for the accretion disk is
assumed a priori, the uncertainties and covariances between the parameters of
the model and the parameters that control the deformation from General
Relativity make any test of General Relativity very challenging with accretion
disk spectrum observations. We also discuss the implications of assuming that
General Relativity is correct a priori on the estimation of parameters of the
astrophysical model when the data is not described by Einstein's theory, which
can lead to a fundamental systematic bias.
