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 definite mechanical effect on a measuring probe specifically designed to minimize the back reaction on the measured system. We then present an interesting 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 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.