Recovering standard methods and short-cuts to Bayesian reasoning

We have already used this example
in Section ,
when we were discussing
the arbitrariness of probability inversion performed
unconsciously by (most of)^{2.13}
those who use the scheme of
confidence intervals. The same example will also be used in
Section , when discussing the reason why Bayesian estimators
appear to be distorted (a topic discussed in more detail in
Section ).
This analogy is very important,
and, in many practical applications, it allows us
to bypass the explicit
use of Bayes' theorem when priors
do not sizably influence the result
(in the case of a normal model the demonstration can be seen
in Section ).

- One is allowed to use these methods if one thinks that the approximations are valid; the same happens with the usual propagation of uncertainties and of their correlations, outlined in the next section.
- One keeps the Bayesian interpretation of the results; in particular, one is allowed to talk about the probability distributions of the true values, with all the philosophical and practical advantages we have seen.
- Even if the priors are
not negligible, but the final distribution
is roughly normal,
^{2.14}one can evaluate the expected value and standard deviation from the shape of the distribution, as is well known:

(2.2) (2.3)

where stands for the mode of the distribution.