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Weak measurement of a quantum system followed by postselection based on asubsequent strong measurement gives rise to a quantity called the weak value: acomplex number for which the interpretation has long been debated. We analysethe procedure of weak measurement and postselection, and the interpretation ofthe associated weak value, using a theory of classical mechanics supplementedby an epistemic restriction that is known to be operationally equivalent to asubtheory of quantum mechanics. Both the real and imaginary components of theweak value appear as phase space displacements in the postselected expectationvalues of the measurement device's position and momentum distributions, and werecover the same displacements as in the quantum case by studying thecorresponding evolution in the classical theory. By using this analogousclassical theory, we gain insight into the appearance of the weak value as aresult of the statistical effects of post selection, and this provides us withan operational interpretation of the weak value, both its real and imaginaryparts. We find that the imaginary part of the weak value is a measure of howmuch postselection biases the mean phase space distribution for a given amountof measurement disturbance. All such biases proportional to the imaginary partof the weak value vanish in the limit where disturbance due to measurement goesto zero. Our analysis also offers intuitive insight into how measurementdisturbance can be minimised and the limits of weak measurement.

Weak values in a classical theory with an epistemic restriction