  We introduce a simple general method for finding the equilibrium distribution
for a class of widely used inexact Markov Chain Monte Carlo algorithms. The
explicit error due to the non-commutivity of the updating operators when
numerically integrating Hamilton's equations can be derived using the
Baker-Campbell-Hausdorff formula. This error is manifest in the conservation of
a ``shadow'' Hamiltonian that lies close to the desired Hamiltonian. The fixed
point distribution of inexact Hybrid algorithms may then be derived taking into
account that the fixed point of the momentum heatbath and that of the molecular
dynamics do not coincide exactly. We perform this derivation for various
inexact algorithms used for lattice QCD calculations.
