  The absence of a missing moment inertia in clean solid $^4$He suggests that
the minimal experimentally relevant model is one in which disorder induces
superfluidity in a bosonic lattice. To this end, we explore the relevance of
the disordered Bose-Hubbard model in this context. We posit that a clean array
$^4$He atoms is a self-generated Mott insulator, that is, the $^4$He atoms
constitute the lattice as well as the `charge carriers'. With this assumption,
we are able to interpret the textbook defect-driven supersolids as excitations
of either the lower or upper Hubbard bands. In the experiments at hand,
disorder induces a closing of the Mott gap through the generation of mid-gap
localized states at the chemical potential. Depending on the magnitude of the
disorder, we find that the destruction of the Mott state takes place for
$d+z>4$ either through a Bose glass phase (strong disorder) or through a direct
transition to a superfluid (weak disorder). For $d+z<4$, disorder is always
relevant. The critical value of the disorder that separates these two regimes
is shown to be a function of the boson filling, interaction and the momentum
cut off. We apply our work to the experimentally observed enhancement $^3$He
impurities has on the onset temperature for the missing moment of inertia. We
find quantitative agreement with experimental trends.
