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# Problem and typical solutions

In any experiment the distribution of the measured observables differs from that of the corresponding true physical quantities due to physics and detector effects. For example, one may be interested in measuring the variables and in deep-inelastic scattering events. In such a case one is able to build statistical estimators which in principle have a physical meaning similar to the true quantities, but which have a non-vanishing variance and are also distorted due to QED and QCD radiative corrections, parton fragmentation, particle decay and limited detector performances. The aim of the experimentalist is to unfold the observed distribution from all these distortions so as to extract the true distribution (see also Refs. [54] and [55]). This requires a satisfactory knowledge of the overall effect of the distortions on the true physical quantity.

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.

Next: Bayes' theorem stated in Up: Bayesian unfolding Previous: Bayesian unfolding   Contents
Giulio D'Agostini 2003-05-15