The results are summarized in
Figure 3 shows clearly how the initial beliefs about (and therefore on ) are updated, within each model. We want to stress that the final conclusion depends still on the prior beliefs. If someone thought that had to be above 10 this person had to reconsider completely his/her beliefs, independently from the model; if another person believed that only values below 0.01 were reasonable, the experiment would not affect at all his/her beliefs, independently of the model. For this reason, the ML value could be misleading if erroneously associated, as it often happens, to the value around which our confidence is finally concentrated, independently from any prior knowledge. Nevertheless, and with these warnings, we report in Tab. 1 also the results obtained from a ML analysis and from a naïve Bayesian inference that assumes a uniform prior on (and therefore on and , since they differ by factors). has been evaluated from the curvature of the minus-log-likelihood around its minimum, i.e. . The results of the `naïve Bayesian inference' are reported as expected values E and standard deviations evaluated from the final distribution. The condition has been written explicitly in E and , according to the Bayesian spirit. Note that, for obvious reasons, the mode of the posterior calculated using a uniform prior is exactly equivalent to the ML estimate. This observation is important to understand the slightly different results obtained with the two methods. The posterior expected value is always larger than the ML one, simply because of the asymmetry of .
Perhaps is the most interesting quantity to understand the conclusions of these model dependent analyzes that, we like to repeat it, do not take properly into account prior knowledge. The three physical model suggest about 10 coincidences due to g.w.'s, with a 50% uncertainty. Instead, for the unphysical model (ISO) less events are found and with larger uncertainty. Note that, for this model, the mode of the posterior (or, equivalently, the estimate) gives a number of candidate events that is the difference between the total number of observed events and that expected from the background alone. Instead, for the three Galactic models, a number of events larger than this difference is attributed to the signal, as a consequence of a `possibly good' time modulation recognized in the data (in other words, the method `likes to think' that, given a time distribution shape that reminds the pattern of the Galactic models, the background has most likely under-fluctuated within what is reasonably allowed by its probability distribution).
To summarize this subsection, the three Galactic models show good agreement in indicating for which values of g.w. events, or event rate, we must increase our beliefs. But the final beliefs depend on our initial ones, as explained introducing the Bayesian approach. If you think that, given your best knowledge of the models of g.w.'s sources and of g.w. interaction with cryogenic detectors, a g.w. rate on Earth of up to event/day is quite possible, the data make you to believe that this rate is event/day and that they contain genuine g.w. coincidences.