It is first demonstrated that every LQG optimal control problem can be rewritten as a minimum output variance control problem for a related system model. This system may be defined in either minimum or non-minimum phase forms. The optimal controller derived from the equivalent minimum variance problem is identical to the LQG optimal controller. During the solution of the equivalent problem an equation is derived which is similar to the implicit relationships used in direct self-tuning control schemes. This enables the problems in constructing a multivariable implicit LQG self-tuning controller, using polynomial matrix methods, to be explored. The inherent difficulties become clear but a possible strategy is evident. Finally the results are specialized to the scalar case and a very simple implicit LQG self-tuning algorithm is described.