  Weak measurement of a quantum system followed by postselection based on a
subsequent strong measurement gives rise to a quantity called the weak value: a
complex number for which the interpretation has long been debated. We analyse
the procedure of weak measurement and postselection, and the interpretation of
the associated weak value, using a theory of classical mechanics supplemented
by an epistemic restriction that is known to be operationally equivalent to a
subtheory of quantum mechanics. Both the real and imaginary components of the
weak value appear as phase space displacements in the postselected expectation
values of the measurement device's position and momentum distributions, and we
recover the same displacements as in the quantum case by studying the
corresponding evolution in the classical theory. By using this analogous
classical theory, we gain insight into the appearance of the weak value as a
result of the statistical effects of post selection, and this provides us with
an operational interpretation of the weak value, both its real and imaginary
parts. We find that the imaginary part of the weak value is a measure of how
much postselection biases the mean phase space distribution for a given amount
of measurement disturbance. All such biases proportional to the imaginary part
of the weak value vanish in the limit where disturbance due to measurement goes
to zero. Our analysis also offers intuitive insight into how measurement
disturbance can be minimised and the limits of weak measurement.
