  We study the maximum likelihood degree of linear concentration models in
algebraic statistics. We relate the geometry of the reciprocal variety to that
of semidefinite programming. We show that the Zariski closure in the
Grassmanian of the set of linear spaces that do not attain their maximal
possible maximum likelihood degree coincides with the Zariski closure of the
set of linear spaces defining a projection with non-closed image of the
positive semidefinite cone. In particular, this shows that this closure is a
union of coisotropic hypersurfaces.
