MMP.7:5.2 - Preserve a common influence across readings

Two sensors measure quantities x1 and x2 with one shared calibration offset b. Their readings are Y1=x1+b+E1 and Y2=x2+b+E2, with independent zero-mean errors of variance sigma squared. Begin by retaining b as a common parameter. The joint conditional density factors given b; each factor uses that same value.

For the difference, Y1-Y2=x1-x2+E1-E2: the offset cancels. Its error variance is 2 sigma^2. A measurement of the difference can therefore be useful while either absolute value remains uncertain.

For repeated measurements of one x, suppose an additional justified model describes the common offset as a zero-mean random variable B with variance tau squared, independent of the errors. The average of n readings has variance tau^2 + sigma^2/n. Integrating a separate offset for every reading would incorrectly produce (tau^2+sigma^2)/n. Repetition reduces independent noise but leaves this common calibration contribution.

If the instrument is independently recalibrated before every reading, the arrangement changes. A separate-offset model can then be appropriate. The governing operation is to trace which influences are shared and preserve that sharing in the probability construction.