  The fluctuation theorem for entropy production is a remarkable symmetry of
the distribution of produced entropy that holds universally in non-equilibrium
steady states with Markovian dynamics. However, in systems with slow degrees of
freedom that are hidden from the observer, it is not possible to infer the
amount of produced entropy exactly. Previous work suggested that a relation
similar to the fluctuation theorem may hold at least approximately for such
systems if one considers an apparent entropy production. By extending the
notion of apparent entropy production to discrete bipartite systems, we
investigate which criteria have to be met for such a modified fluctuation
theorem to hold in the large deviation limit. We use asymptotic approximations
of the large deviation function to show that the probabilities of extreme
events of apparent entropy production always obey a modified fluctuation
theorem and, moreover, that it is possible to infer otherwise hidden
properties. For the paradigmatic case of two coupled colloidal particles on
rings the rate function of the apparent entropy production is calculated to
illustrate this asymptotic behavior and to show that the modified fluctuation
theorem observed experimentally for short observation times does not persist in
the long time limit.
