Problem and typical solutions

When dealing with only one physical variable the usual method
for handling this problem is the so-called
*bin-to-bin* correction: one evaluates
a *generalized efficiency* (it
may even be larger than unity) by
calculating the ratio between the number of events falling in a certain
bin of the reconstructed variable and the number of events in the
__same__ bin of the true variable
with a Monte Carlo simulation.
This efficiency is then used
to estimate the number of true events from the
number of events observed in that bin. Clearly this method requires the
same subdivision in bins of the true and the experimental variable
and hence it cannot take into account
large migrations of events from one
bin to the others.
Moreover it neglects the
unavoidable correlations between adjacent bins. This approximation is valid
only if the amount of migration is negligible
and if the standard deviation of the
smearing is smaller than the bin size.

An attempt to solve the problem of migrations is sometimes made by building a matrix which connects the number of events generated in one bin to the number of events observed in the other bins. This matrix is then inverted and applied to the measured distribution. This immediately produces inversion problems if the matrix is singular. On the other hand, there is no reason from a probabilistic point of view why the inverse matrix should exist. This can easily be seen by taking the example of two bins of the true quantity both of which have the same probability of being observed in each of the bins of the measured quantity. It follows that treating probability distributions as vectors in space is not correct, even in principle. Moreover the method is not able to handle large statistical fluctuations even if the matrix can be inverted (if we have, for example, a very large number of events with which to estimate its elements and we choose the binning in such a way as to make the matrix not singular). The easiest way to see this is to think of the unavoidable negative terms of the inverse of the matrix which in some extreme cases may yield negative numbers of unfolded events. Quite apart from these theoretical reservations, the actual experience of those who have used this method is rather discouraging, the results being highly unstable.