Abstract
In order to measure something, we have first to construct a reference scale for the quantity of interest. This may be done, at least in principle, by selecting a subset of objects which is representative of all the possible states of the quantity and by assigning them a measure, in order to constitute a reference scale.
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Notes
- 1.
Note that the function \(m\) does not represent an empirical operation, as \(\gamma \) instead does, but rather the (mathematical) existence of a correspondence between objects and numbers, as ensured by the representation theorems that we have discussed in Chap. 3. In performing measurement , therefore, \(x\) has to be regarded as an unknown value, whilst \(\hat{x}\) is the value that we actually obtain as the result of the measurement process.
- 2.
Calibration will be treated in greater detail in Chap. 11.
- 3.
Note that here \(m\) denotes the “measure” function, amply discussed in Chap. 3, and not “mass”, as in the previous example.
- 4.
Restitution may be sometimes very simple, as in the case of direct measurement, where it just consists in assigning to the measurand the same value of the standard that has been recognised as equivalent to it. In other cases, instead, it may be very complicated and challenging, as it happens in image-based measurement, where it involves sophisticated image processing procedures. Anyway, it is conceptually always present, since the instrument indication is, in general, a sign that needs to be interpreted for obtaining the measurement value, and restitution constitutes such an interpretation.
- 5.
Calibration will be treated in some detail in Chap. 11.
- 6.
This substitution may sound odd to some readers familiar with Bayesian statistics. The reason for this substitution is that \(x\) and \(x^{\prime }\) describe what happens in two distinct stages in the measurement process. Additional reasons for this distinction will appear in the next sections and in Chap. 6.
- 7.
This factorisation in three distributions simply results from the application of a rule of probability calculus; see e.g. [8].
- 8.
If instead, for some \(x\), \(P(y|x)=P(y)\), the indication would be, in those cases, independent from \(x\), and measurement would be thus impossible, since we would not obtain any information from the instrument.
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Rossi, G.B. (2014). The Measurement Process. In: Measurement and Probability. Springer Series in Measurement Science and Technology. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8825-0_5
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