Next: Combination of several measurements Up: Normally distributed observables Previous: Normally distributed observables   Contents

## 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 variance5.2.) 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:
 (5.12) (5.13)

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 2003-05-15