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Combination of results: general case

The previous case is rather artificial and can be used, at most, to combine several measurements of the same experiment repeated times, each with the same running time. In general, experiments differ in size, efficiency, and running time. A result on is no longer meaningful. The quantity which is independent from these contingent factors is the rate, related to by

where indicates the efficiency, the generic size' (either area or volume, depending on whatever is relevant for the kind of detection) and the running time: all the factors have been grouped into a generic integrated luminosity' which quantify the effective exposure of the experiment.

As seen in the previous case, the combined result can be achieved using Bayes' theorem iteratively, but now one has to pay attention to the fact that:

• the observable is Poisson distributed, and the each experiment can infer a parameter;
• the result on must be translated9.2into a result on .
Starting from a prior on (e.g. a monopole flux) and going from experiment 1 to we have
• from and we get ; then, from the data we perform the inference on and then on :
 Data Data Data

• The process is iterated for the second experiment:

• and so on for all the experiments.
Lets us see in detail the case of null observation in all experiments ( ) , starting from a uniform distribution.
Experiment 1:

 (9.7) at 95% probability (9.8)

Experiment 2:

Experiment :

 (9.9)

The final result is insensitive to the data grouping. As the intuition suggests, many experiments give the same result of a single experiment with equivalent luminosity. To get the upper limit, we calculate, as usual, the cumulative distribution and require a certain probability for to be below [i.e. ]:

obtaining the following rule for the combination of upper limits on rates:

 (9.10)

We have considered here only the case in which no background is expected, but it is not difficult to take background into account, following what has been said in Section .

Next: Including systematic effects Up: Poisson model: dependence on Previous: Combination of results from   Contents
Giulio D'Agostini 2003-05-15