  We introduce a simple geometrical construction similar to covariant
holographic entanglement entropy but with the addition of a new term
proportional to boundary region volume. This new procedure has properties
strongly resembling classical Shannon entropy of probability distributions
rather than von Neumann entropy, so we call the quantity holographic Shannon
entropy. The holographic Shannon entropy of a region $A$ is divergent in
AdS/CFT, but upon regulation, it appears to be equal to the outer entropy of
the entanglement wedge of $A$ with the entanglement wedge of the complement of
$A$ held fixed. The construction is unambiguous when applied to compact
surfaces with convex shape in general spacetimes obeying the null curvature
condition (of which AdS with a regulator is a special case). In this context we
prove that holographic Shannon entropy obeys monotonicity, a key property of
Shannon entropy, as well as all known balanced inequalities of dynamical
holographic entanglement entropy. In the static case, we explain why there must
exist some classical probability distribution on random variables locally
distributed on the boundary with the property that the Shannon entropies of all
marginals are exactly reproduced by the holographic Shannon entropy formula.
