While in the previous subsection we have been interested to learn about or within a model (and then, since all results are conditioned by that model, it makes no sense from that perspective to state if the model is right or wrong), let us see now how to modify our beliefs on each model. This is a delicate question to be treated with care. Intuitively, we can imagine that we have to make use of the values, in the sense that the higher is the value and the most `the hypothesis' increases its credibility. The crucial point is to understand that `the hypothesis' is, indeed, a complex (somewhat multidimensional) hypothesis. Another important point is that, given a non null background and the properties of the Poisson distribution, we are never certain that the observations are not due to background alone (this is the reason why the function does not vanish for ).
The first point can be well understood making an example based
on Fig. 3 and Tab. 1.
for the different models one could come
to the rash conclusion that the Galactic Center model is enhanced by
21 with respect to the non g.w. hypothesis, or that
the Galactic Center model is enhanced by a factor
with respect to the hypothesis of signals
from sources uniformly distributed over the Galactic Disk.
However these conclusions would be correct only in the case that each
model would admit only that value of the parameter which
maximizes , i.e.
Let us take the Bayes factor defined in Eq. (7). The probability theory teaches us promptly what to do when each model depends on parameters:
To better understand the role of the parameter prior in Eq. (21), let us take the example of a model (which we do not consider realistic and, hence, we have discarded a priori in our analysis) that gives a signal only in one of the 1/2 hours bins, being all bins a priori equally possible. This model would depend on two parameters, and , where is the center of the time bin. Considering and independent, the parameter prior is , where is a probability function for the discrete variable . The `evidence' for this model would be
As we have seen, while the Bayes factors for simple hypotheses (`simple' in the sense that they have no internal parameters) provide a prior-free information of how to modify the beliefs, in the case of models with free parameters Bayes factors remain independent from the beliefs about the models, but do depend on the priors about the model parameters. In our case they depend on the priors about , which might be different for different models. If we were comparing different models, each with its about which there is full agreement in the scientific community, all further calculations would be straightforward. However, we do not think to be in such a nice text-book situation, dealing with open problems in frontier physics (for example, note that , and then and , depend on the g.w. cross section on cryogenic bars, and we do not believe that the understanding of the underlying mechanisms is completely settled). In principle every physicist which have formed his/her ideas about some model and its parameters should insert his/her functions in the formulae and see from the result how he/she should change his/her opinion about the different models. Virtually our task ends here, having given the functions, which can be seen as the best summary of an experimental fact, and having indicated how to proceed (for recent examples of applications of this method in astrophysics and cosmology see Refs. [10,11,12]). Indeed, we proceed, showing how beliefs can change given some possible scenarios for .
The first scenario is that in which the possible value of are considered so small that is equal to zero for . The result is simple: the data are irrelevant and beliefs on the different models are not updated by the data.
Other scenarios might allow
the possibility that
is positive for values up to
and more. We shall use
three different pdf's for as examples of prior beliefs,
that we call `sceptical', `moderate' and 'uniform' (up to ).
The `moderate' pdf corresponds to a rate
which is rapidly going to zero around the value which we have measured.
The initial pdf is modeled with a half-Gaussian with .
The `sceptical' pdf has a ten times smaller.
The `uniform' considers equally likely
all up to the last decade
in which the functions are sizable different from zero.
Here are the three
Using these three pdf's for the parameter ,
we can finally calculate all Bayes factors.
We report in Tab. 2 the Bayes factors of the models
of Fig. 2 with respect to model
using Eq. (20).
All other Bayes factors can be calculated as ratio of these.
We have also considered a prior which is uniform in , between . This prior accords equal probability to each decade in the parameter , and probably accords many people prior intuition. Bayes factors, for the four models of Fig. 2 with respect to model ``only background'', are:
4.0 (GC); 2.0 (GD); 2.2 (GMD); 1.0 (ISO).
Again, within this scenario there is some preference for the Galactic Center model with a Bayes factor about 2 with respect to each other model.