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Inferring and

The histograms of Fig. 11 show examples of the probability distributions of for and three different hypotheses for .
These distributions quantify how much we believe that out of the observed belong to the signal. [By the way, the number of background objects present in the data can be inferred as complement to , since the two numbers are linearly dependent. It follows that .]

A different question is to infer the the Poisson of the signal. Using once more Bayes theorem we get, under the hypothesis of signal objects:

 (45)

Assuming a uniform prior for we get (see e.g. Ref. [2]):
 (46)

with expected value and variance both equal to and mode equal to (the expected value is shifted on the right side of the mode because the distribution is skewed to the right). Figure 12 shows these pdf's, for ranging from 0 to 12 and assuming a uniform prior for .

As far the pdf of that depends on all possible values of , each with is probability, is concerned, we get from probability theory [and remembering that, indeed, is equal to , because depends only on , and then the other way around]:

 (47)

i.e. the pdf of is the weighted average2 of the several depending pdf's.

The results for the example we are considering in this section are given in the plots of Fig. 11.

Next: Conclusions Up: Poisson background on the Previous: Inferring
Giulio D'Agostini 2004-12-13