Systematic errors

For each coordinate we can introduce the fictitious quantities
and that take into account the
modification of and due to the systematic effect.
For example, if the systematic effects only acts as an *offset*,
i.e. we are uncertain about the
`true' *zero* of the instruments, and ,
we have

(71) | |||

(72) |

where the true value of are unknown (otherwise there would be no systematic errors). We only know that their expected value is zero (otherwise we need to apply a calibration constant to the measurements) and we quantify our uncertainty with pdf's. For example, we could model them with Gaussian distributions:

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(74) |

Anyway, for sake of generality, we leave the systematic effects in the most general form, dependent on the uncertain quantities and [to be clear: in the case of solely offset systematics we have ]. The values of and are modeled as follow

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(78) |

Figure 3 shows the graphical model containing the new ingredients. The links and are to remember that systematics could also effect the error functions. An alternative visual picture of the probabilistic model is shown in Fig. 4. Note the different symbols to indicate the different uncertain processes: the divergent arrows (in yellow, if you are reading an electronic version of the paper) indicate that, given a value of the `parent' variable, the `child' variable fluctuates on an event-by-event basis; the green single arrow with the question mark indicate that, given a value of the `parent', the child will always take a fixed value, though we do not know which one.

Obviously, the practical implementation of complicate systematic
effects in complicate fits can be quite challenging, but at least
the Bayesian network provides an overall picture of the model.
The simplest case is that of linear fit where only offset
and scale uncertainty are present, with uncertainty modeled
by a Gaussian distribution.
This means that the
's and their uncertainty are as follows
( is the scale factor of uncertain value):

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(80) | |||

(81) |

In this case we can get an hint of how the uncertainty about and change without doing the full calculation following an heuristic approach, valid when is approximately multivariate Gaussian and the details of which can be found in Ref. [16]. We obtain the following results, in which indicates the contribution to the uncertainty about the slope due to uncertainty about , that due to the scale factor , and so on

(82) | |||

(83) | |||

(84) | |||

(85) | |||

(86) | |||

(87) | |||

(88) | |||

(89) |

All contributions are then added quadratically to the so called `statistical' ones.