Extra variability of the data

More rigorously, this formula can be obtained from a variation of reasoning followed in the previous section.

- depends on and on the set of hidden
variables
:

(35) (36)

where the overall dependence has been split in two functions: , only depending on and the model parameters, corresponding to the ideal case; describing the difference from the ideal case. - Calling the fictitious variable, deterministically dependent
on , for a given we have the following model

(37) (38)

where describes our uncertainty about due to the unknown values of all other hidden variables. - We need now to specify
. As usual,
in lack of better knowledge, we take a Gaussian distribution
of unknown parameter ,
with awareness that this is just a convenient, approximated
way to quantify our uncertainty.
At this point a summary of all ingredients of the model in the specific case of linear model is in order:

where stands for a uniform distribution over a very large interval, and the symbol `' has been used to deterministically assign a value, as done in BUGS[11] (see later). - We have now the extra parameter that we include in
,
so that increases by 1.
The new model in represented in Fig. 2,
**Figure:***Minimal modification of Fig. 1 to model the extra variability not described by the error functions. Note that stands for all model parameters to be inferred, including . Instead, stands for all parameters apart from .* - The variables of the model are now , and Eq. (22)
becomes

- Consequently, Eq. (10) becomes

- Inserting the model functions (40)-(45) in
Eq. (46), after the marginalization
(47) and the factorization of the result
into likelihood as prior
[as previously done in (24)], we get
the analogues of Eqs. (26)-(28):

- Inserting in Eq. (25) the expression of coming from Eq. (52) we get finally Eq. (35).