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Final distribution, prevision and credibility intervals of
the true value
The first application of the Bayesian inference will be that
of a normally distributed quantity. Let us take
a data sample
of measurements, of which
we calculate the average
. In our formalism
is a realization of the random variable
. Let us assume we know the
standard deviation of the variable , either
because is very large and can be estimated
accurately from the sample or because it was known a priori
(We are not going to discuss in these notes the case
of small samples and unknown variance^{5.2}.) The property of the average (see Section )
tells us that the
likelihood
is Gaussian:

(5.11) 
To simplify the following notation, let us call
this average and the standard deviation of the average:
We then apply () and get

(5.14) 
At this point we have to make a choice for
. A reasonable choice
is to take, as a first guess,
a uniform distribution defined over a ``large''
interval which includes . It is not really important
how large the interval is,
for a few away
from the integrand at the denominator
tends to zero because of the Gaussian function. What is important
is that a constant
can be simplified
in (), obtaining

(5.15) 
The integral in the denominator is equal to unity, since
integrating with
respect to is equivalent to integrating with respect to .
The final result is then

(5.16) 
 the true value is normally distributed around ;
 its best estimate (prevision) is
E;
 its variance is
;
 the ``confidence intervals'', or credibility intervals,
in which there is a certain probability of finding the
true value are easily calculable:
Probability level 
credibility interval 
(confidence level) 
(confidence interval) 


68.3 



90.0 



95.0 



99.0 



99.73 



Next: Combination of several measurements
Up: Normally distributed observables
Previous: Normally distributed observables
Contents
Giulio D'Agostini
20030515