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##

Hierarchical modelling and hyperparameters

As we have seen in the previous section,
it is often desirable to include in a probabilistic model one's
uncertainty in various aspects of a pdf.
This is a natural feature of the Bayesian methods,
due to the uniform approach to deal with uncertainty
and from which powerful analysis tools are derived.
This kind of this modelling is called
*hierarchical* because the characteristics
of one pdf are controlled by another pdf.
All uncertain parameters from which the pdf
depends are called *hyperparameter*.
An example of use of hyperparameter is described in
Sect. 8.3 in which the
prior to infer in a binomial model are shown to be controlled
by the parameters of a Beta distribution.
As an example of practical importance, think of
the combination of experimental results in the presence
of *outliers*, i.e. of data points which are
somehow in mutual disagreement. In this case the combination
rule given by Eqs. (30)-(32),
extended to many data points, produces unacceptable conclusions.
A way of solving the problem (Dose and von der Linden 1999,
D'Agostini 1999b) is to model a scepticism about the quoted
standard deviations of the experiments, introducing
a pdf , where is a rescaling factor of the
standard deviation. In this way the 's that enter
the r.h.s. of Eqs. (30)-(32)
are hyperparameters of the problem.
An alternative approach,
also based on hierarchical modelling, is shown in (Fröhner 2000).
For a more complete introduction to the subject see e.g.
(Gelman *et al *1995).

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Giulio D'Agostini
2003-05-13