  Strongly interacting systems can be described in terms of correlation
functions at various orders. A quantum analog of high-order correlations is the
topological entanglement in topologically ordered states of matter at zero
temperature, usually quantified by topological entanglement entropy (TEE). In
this work, we propose a statistical interpretation that unifies the two under
the same information-theoretic framework. We demonstrate that the existence of
a non-zero TEE can be understood in the statistical view as the emergent $n$th
order mutual information $I_n$ (for arbitrary integer $n\ge 3$) reflected in
projectively measured samples, which also makes explicit the equivalence
between the two existing methods for its extraction -- the Kitaev-Preskill and
the Levin-Wen construction. To exploit the statistical nature of $I_n$, we
construct a restricted Boltzmann machine (RBM) which captures the high-order
correlations and correspondingly the topological entanglement that are encoded
in the distribution of projected samples by representing the entanglement
Hamiltonian of a local region under the proper basis. Furthermore, we derive a
closed form which presents a method to interrogate the trained RBM, making
explicit the analytical form of arbitrary order of correlations relevant for
$I_n$. We remark that the interrogation method for extracting high-order
correlation can also be applied to the construction of auxiliary fields that
disentangle many-body interactions relevant for diverse interacting models.

Quantum Physics;Disordered Systems and Neural Networks;Strongly Correlated Electrons