  We re-examine the status of the weak value of a quantum mechanical observable
as an objective physical concept, addressing its physical interpretation and
general domain of applicability. We show that the weak value can be regarded as
a \emph{definite} mechanical effect on a measuring probe specifically designed
to minimize the back-reaction on the measured system. We then present a new
framework for general measurement conditions (where the back-reaction on the
system may not be negligible) in which the measurement outcomes can still be
interpreted as \emph{quantum averages of weak values}. We show that in the
classical limit, there is a direct correspondence between quantum averages of
weak values and posterior expectation values of classical dynamical properties
according to the classical inference framework.
