  We expand our recent work on the outer entropy, a holographic coarse-grained
entropy defined by maximizing the boundary entropy while fixing the classical
bulk data outside some surface. When the surface is marginally trapped and
satisfies certain "minimar" conditions, we prove that the outer entropy is
exactly equal to a quarter the area (while for other classes of surfaces, the
area gives an upper or lower bound). We explicitly construct the
entropy-maximizing interior of a minimar surface, and show that it satisfies
the appropriate junction conditions. This provides a statistical explanation
for the area-increase law for spacelike holographic screens foliated by minimar
surfaces. Our construction also provides an interpretation of the area for a
class of non-minimal extremal surfaces.
  On the boundary side, we define an increasing simple entropy by maximizing
the entropy subject to a set of "simple experiments" performed after some time.
We show (to all orders in perturbation theory around equilibrium) that the
simple entropy is the boundary dual to our bulk construction.
