  We give a new derivation of the variational formula for the pressure of the
long-range-hopping Bose-Hubbard model, which was first proved in \cite{BD}. The
proof is analogous to that of a theorem on noncommutative large deviations
introduced by Petz, Raggio and Verbeure \cite{PRV} and could similarly be
extended to more general Bose system of mean-field type. We apply this
formalism to prove Bose-Einstein condensation for the case of small coupling.
