The Bayesian way out: how to use of experimental data to update the credibility of hypotheses

where stand the hypotheses that could produce the

The presence of priors, considered a weak point by opposer's of the Bayesian theory, is one of the points of force of the theory. First, because priors are necessary to make the `probability inversion' of Eq. (4). Second, because in this approach all relevant conditions must be clearly stated, instead of being hidden in the method or left to the arbitrariness of the practitioner. Third, because prior knowledge can be properly incorporated in the analysis to integrate missing or deteriorated experimental information (and whatever it is done should be stated explicitly!). Finally, because the clear separation of prior and likelihood in Eq. (4) allows to publish the results in a way independent from , if the priors might differ largely within the members of the scientific community. In particular, the Bayes factor, defined as

is the factor which changes the `bet odds' (i.e. probability ratios) in the light of the new data. In fact, dividing member to member Eq. (4) written for hypotheses and , we get

Since we shall speak later about models , the odd ratio updating is given by

Some general remarks are in order.

- Conclusions depend only on the observed data and on the previous knowledge. In particular they do not depend on unobserved data which are rarer than the data really observed (that is what p-values imply).
- At least two models have to be taken into account, and the likelihood for each model must be specified.
- There is no need to consider `all possible models', since what matters are relative beliefs.
- Similarly, there is no need that the model must be declared before the data
are taken, or analyzed.
What matters is that the
*initial*beliefs should be based on general arguments about the plausibility of each model and on agreement with other experimental information, other than*Data*.

where stand for probability density functions (pdf) Also in this case, a prior independent way of reporting the result is possible. The difficulty of dealing with an infinite number of Bayes factors (precisely , given each and ) can be overcome defining a function of which gives the Bayes factor with respect to a reference . This function is particularly useful if is chosen to be the asymptotic value at which the experiment looses completely sensitivity. For g.w. search this asymptotic value is simply . In other cases it could be an infinite particle mass [3] or an infinite mass scale [4]. In the case of g.w. rate , extensively discussed in Ref. [5], we get

where is the model dependent likelihood. [Note that, indeed, in the limit of the likelihood depends only on the background expectation and not on the specific model. Therefore , where stands for the model ``background alone''.] This function has the meaning of

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- Models for Galactic sources of gravitational waves
- Relative belief updating factor for a given model
- Model comparison taking into account the a priori possible values of the model parameters