  A model is proposed for the statistical analysis of arbitrary-strength
quantum measurements, based on a picture of "sampling weak values" from
different configurations of the system. The model is comprised of two elements:
a "local weak value" and a "likelihood factor". The first describes the
response of an idealized weak measurement situation where the back-reaction on
the system is perfectly controlled. The second assigns a weight factor to
possible configurations of the system. The distribution of the data in a
measurement of arbitrary strength may the be viewed as the net result of
interfering different samples weighted by the likelihood factor, each of which
implements a weak measurement of a different local weak value. It is shown that
the mean and variance of the data can be connected directly to the means and
variances of the sampled weak values. The model is then applied to a situation
similar to a phase transition, where the distribution of the data exhibits two
qualitatively different shapes as the strength parameter is slightly varied
away from a critical value: one below the critical point, where an unusual weak
value is resolved, the other above the critical point, where the spectrum of
the measured observable is resolved. In the picture of sampling, the transition
corresponds to a qualitative change in the sampling profile brought about by
the competition between the prior sampling distribution and the likelihood
factor.
