We can easily extend Eqs. (73),
(77), and (79)
to a joint inference of several variables, which, as we
have seen, are nothing but parameters
of suitable models. Using the alternative ways described in
Sects. 6.1 and 6.2, we have

and

respectively. The two ways lead to an identical result, as it can be seen comparing Eqs. (81) and (83).

Take a simple case of a common offset error of an instrument used to measure various quantities , resulting in the measurements . We model each measurement as plus an error that is Gaussian distributed with a mean of zero and a standard deviation . The calculation of the posterior distribution can be performed analytically, with the following results (see D'Agostini 1999c for details):

- The uncertainty in each is described by a Gaussian centered at , with standard deviation , consistent with Eq. (76).
- The joint posterior distribution does not factorize into the product of , , etc., because correlations are automatically introduced by the formalism, consistent with the intuitive thinking of what a common systematic should do. Therefore, the joint distribution will be a multi-variate Gaussian that takes into account correlation terms.
- The
*correlation coefficient*between any pair is given by

(84)

We see that has the behavior expected from a common offset error; it is non-negative; it varies from practically zero, indicating negligible correlation, when ( ), to unity ( ), when the offset error dominates.