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# Extra variability of the data

As clearly stated, the previous results assume that the only sources of deviation of the measurements from the value of the physical quantities are normal errors, with known standard deviations and . Sometimes, as it is the case of the data points reported in Ref. [14], this is not the case. This means that depends also on other, hidden' variables, and what we observe is the overall effects integrated over all the variability of the variables that we do not see'. In lack of more detailed information, the simplest modification to the model described above is to add an extra Gaussian noise' on one of the coordinates. For tradition and simplicity this extra noise is added to the variable. The effect on the above result can be easily understood. Let us call the r.m.s. of this extra noise that acts normally and independently in each point. As it is well known, the sum of Gaussian distributions is still Gaussian with an expected value and variance respectively sum of the individual expected values and variances. Therefore, the effect in the individual likelihoods (28) is to replace by . But we now have an extra parameter in the model, and Eq. (30) becomes

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:

 (39) (40) (41) (42) (43) (44)

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,
in which we have indicated by all parameters apart from .
• The variables of the model are now , and Eq. (22) becomes
 (45)

• Consequently, Eq. (10) becomes
 (46)

• 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):
 (47) (48) (49) (50) (51)

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

Next: Computational issues: normalization, fit Up: Fits, and especially linear Previous: Approximated solution for non-linear
Giulio D'Agostini 2005-11-21