As mentioned above, the MRA does not provide a univocal interpretation of the con- cept “equivalence of measurement standards”, but introduces a quantitative measure of equivalence as a certain estimate and corresponding uncertainty, without any expla- nation of what kind of quantity is meant. Naturally, this provokes discussion as well as the wish to determine the degree of equivalence with the help of some particular quantity using an initial model of measurement result:xi DxCbi C"i.
In a number of works the degrees of equivalence are identified with the systematic biases of the measurement results obtained in a respective laboratory. Such considera- tions are based on the model (1.2). Hence, as was noted above, they demand additional assumptions at a simultaneous evaluation of a measurand and systematic biases.
We do not agree with the idea of identifying the degree of measurement standard equivalence with a bias of a measurement result. First of all, let us note that the equiv- alence of measurement standards is established on the basis of a particular set of stan- dards. This is obviously shown above when using a mixture of distributions for a quan- titative expression of the equivalence of standards. As to the bias of a measurement result of some laboratory, it is an intrinsic characteristic of the measurement results obtained in this particular laboratory.
However, it should be recognized that the problem of revealing and evaluating sys- tematic biases in results of comparison participants is rather important. Key compar- isons of national measurement standards provide getting new information about sys- tematic biases of results obtained by laboratories, which is not taken into account in the traditional approach to evaluating comparison data. In this section the problem of evaluating the systematic biases of laboratory results on the basis of data obtained in key comparison of measurement standards is treated without reference to the calcula- tion of the degrees of equivalence.
The application of the Bayesian theorem has allowed a posteriori estimates of sys- tematic biases of laboratories to be obtained without any additional constrains related to the parameters evaluated. Bayesian methods were formerly also applied when eval- uating key comparisons data, but there were difficulties connected with constructing a priori densities of distributions for the parameters being evaluated. These difficulties
have been successfully overcome, using information contained in a budget of uncer- tainty, which is easy to access in each comparison [109, 310].
Let model (1.2) be considered over again:
Xi DXCBi, i D1,: : :,n,
whereXis the measurand that is measured by all comparison participants and remains to be unchangeable at a stable traveling standard, andBi is the systematic bias of a measurement result of a laboratory.
Before carrying out a comparison, some a priori information is available:
the characteristic (standard deviation) of the measurement precision in a laboratory i, which is assumed to be known as an accurate one;
the combined standard uncertainty caused by systematic effectsuB.xi/. It is as- sumed that all known corrections have been already inserted and that the corre- sponding uncertainties of these corrections have been taken into account;
the combined standard uncertainty, correspondingly, equal to u .xi/Dq
uB2 .xi/Ci2.
Thus, before carrying out the comparisons there is no information about a measurand, and with respect to a systematic bias it is possible to tell only that its best interlabo- ratory estimate is equal to zero, with the corresponding uncertaintyuB.xi/ D uBi. This can be formalized in the form of a priori probability density functions (PDFs) of the parameters being evaluated in the following manner:
p .bi/DpBi.tjuBi/D 1
p2uB.xi/exp
² t2
2uB2 .xi/
³
, p.X //1.
On the basis of a priori information and measurement resultsxi, obtained in a key comparison, in each laboratory a posteriori distribution density function of measurand values can be obtained. The application of the Bayesian theorem leads to the following expression for the a posteriori PDF of the measurand:
p.xjxi// Z
l.x,bijxi/p.x,bi/dbi,
where l.x,bijxi/ / expŒ.xi .x C bi//2=2i2 is the likelihood function, and p.x,bi/is the joint PDF of measurand and bias in thei-th laboratory, in the case being consideredp.x,bi//p.bi/.
Before joint handling of all data presented by the comparison participants, it is nec- essary to make sure that the data is consistent. To do this it is possible, for example, to use a known criterion, based on the analysis of the zone of overlapping the PDFs of measurand, obtained in different laboratories [498]. Let us pay attention once again to the fact that model (1.2) is applied in this case for an analysis of consistent data.
Section 1.6 Bayesian approach to the evaluation of systematic biases 29 The application of the Bayesian theorem yields a joint a posteriori PDF of the mea- surand and biases of laboratory results:
p.x,b1,: : :,bNjx1,: : :,xN//
l.x,b1,: : :,bNjx1,: : :,xN/p.x,b1,: : :,bN/, (1.14) where l.x,b1,: : :,bNjx1,: : :,xN/ D l.x,b1jx1/ l.x,bNjxN/ is the likelihood function, andp.x,b1,: : :,bN/ / p.b1/ p.bN/is the PDF of measurand and sys- tematic biases of laboratory results.
Integrating (1.14) yields a posteriori marginal PDFs of the measurand and of the systematic bias in a particular laboratory:
p.xjx1,: : :,xN/D Z
p.x,b1,: : :,bNjx1,: : :,xN/db1 dbN, p.bijx1,: : :,xN/D
Z
p.x,b1,: : :,bNjx1,: : :,xN/dxdb1 dbi1dbiC1 dbN. Let us note that methods of statistic modeling can be used to obtain the a posteriori PDFs of the measurand and the systematic biases for the a priori PDEs of an arbitrary form and for an arbitrary likelihood function. For Gaussian a priori PDFs of systematic biases and the Gaussian likelihood function the joint a posteriori density of distribution is given by the expression
p.x,b1,: : :,bnj: : :,xi, ,uBi: : :.i: : :.// Y
i
p1
2i exp
²
.xixbi/2 2i2
³ 1
p2uBi exp
² bi2
2uBi2
³
. (1.15) The best estimates of a measurand, calculated as a mathematical expectation of the distribution p.xjx1,: : :,xN/, and of the corresponding standard uncertainty, calcu- lated as a standard deviation, are equal to
xw Du2.xw/ XN iD1
xi
.uBi2 Ci2/, u2.xw/D XN
iD1
1 .uBi2 Ci2/
1
. Thus, the Bayesian approach gives a weighted mean of measurement results as an estimate of the measurand. This coincides with the estimate obtained with the least- squares method.
In the same way estimates of systematic biases and corresponding uncertainties have been obtained:
bi D u2Bi
uBi2 Ci2.xixw/, u2.bi/DuBi2
i2Cu2Bi u2.xw/
i2CuBi2
i2CuBi2 . An analysis of the estimates obtained for systematic biases shows that
if the precision of measurements is highi uBi, then the estimate of a systematic bias is close to the difference between the measurement result and weighted mean bi .xixw/. At the same time, the corresponding standard uncertainty is equal tou2.bi/ u2.xw/, i.e., significantly less than the uncertainty obtained for the degree of equivalence;
if a random component is significant, then the role of a priori estimate of a systematic bias in this case becomes more important andjbij jxixwj.
The problem of the evaluation of a systematic difference between the results ob- tained in two laboratories is of independent interest [110, 309]. Based on the model for thei-th laboratory, it is easy to obtain a model of the systematic difference of the results of two laboratories which is equal to the difference of the systematic biases of the results of each laboratory:
XiXj DBiBj.
It is interesting to note that the estimate of the systematic difference between two results depends of the fact of whether the information from all the laboratories or only from those between which the systematic discrepancy of results is determined has been used.
The joint PDF of systematic biases of thei.th andj-th laboratories can be obtained by marginalization of (1.15):
p
bi,bjj: : :,xi, ,uBi: : :.i: : :.
• Y /
i
p 1
2uBi exp
² bi2
2uBi2
³ 1 p2i exp
²
X.xixbi/2 2i2
³
dx db1 dbn
i¤j .
The mathematical expectation of the difference of quantities is equal to the differ- ence of the mathematical expectations:
ij Dbibj D u2Bi
u2BiCi2.xixref/ uBj2
u2Bj Cj2.xj xref/.
To calculate the corresponding uncertainty it is necessary to know the covariation cov.bi,bj/, which can be obtained from the joint probability density function (1.15).
As a result the following is obtained:
u2 ij
D uBi2 i2
uBi2 Ci2C uBj2 j2 uBj2 Cj2 C
´ uBi2
uBi2 Ci2 u2Bj uBj2 Cj2
μ2
u2.xw/. It should be noted that estimates of the systematic difference and associated uncer- tainty obtained using information from all comparison participants differs from the corresponding estimates obtained on the basis of data from only two laboratories. The Bayesian approach on the basis of data from only two laboratories gives the following
Section 1.7 Evaluation of measurement results at calibration 31 estimates of systematic differences:
Qij D
xixj uBi2 CuBj2 i2Cj2CuBi2 CuBj2 , u2Qij
D
i2Cj2 u2BiCuBj2 i2Cj2CuBi2 CuBj2 .
The comparison of two estimates of pair-wise systematic differences shows that
estimates obtained on the basis of information from all participants always have a lesser uncertainty, and the following inequality is true:
u2Qij u2
ij D
uBi2
u2.xi/ uBj2 u2.xj/
!2
8ˆ ˆˆ
<
ˆˆ ˆ:
1 1
u2.xi/C 1 u2
xj
1
X
i
1 u2.xi/
9>
>>
=
>>
>;
>0;
estimates and the corresponding uncertainties coincide, provided that the laborato- ries have similar characteristics of precision and accuracyi D, uBi Du, 8i;
estimates become close to each other at a high precision of measurement results in laboratoriesui
Bi !0.
In Section 1.6 the problem of systematic bias estimation on the basis of the measure- ment results obtained in key comparisons has been considered. The approach applied implies that detailed information about uncertainty components is available. This ap- proach is based on the Bayesian analysis of information from all participants, provided that the consistency of this information is present. As a result, the uncertainties asso- ciated with laboratory bias estimates can be reduced compared to the assessment of a particular laboratory.