4.2. Best estimates and confidence limits

The value y and its combined standard uncertainty u(y) obtained from the evaluation of the output quantity define the mean and the standard deviation of a Gaussian distribution of possible values \(y^{'},\)

\(\frac{1}{u(y)\sqrt{2\pi}}\exp\left( - \frac{\left( y^{'} - y \right)^{2}}{2\ u^{2}(y)} \right)\)

which is obtained from the application of the maximum information entropy principle.

Dependent on the size of the ratio u(y)/y it may happen that a part of this distribution resides in the negative domain (\(y^{'}\)< 0). According to the Bayesian method it follows by using a positive “model prior“, that this distribution function shall only have positive values for positive \(y^{'}\) values (\(y^{'}\): e.g. activity values). Therefore, the part of the distribution in the negative domain is cut at \(y^{'}\)= 0. Using this modified (cut) distribution expectation values of mean and standard deviation can be determined which are here called “best estimates according to Bayes”. Let ω be the value of the integral of the modified distribution function from zero to infinity, with \(\omega < 1\). The resulting expectation values then are:

mean:

\(\widehat{y} = y + \frac{u(y)\ exp\left( - \frac{y^{2}}{2\ u^{2}(y)} \right)}{\omega\sqrt{2\pi}}\) with \(\omega = \Phi\left( \frac{y}{u(y)} \right)\)

(\(\Phi\) is the (cumulative) distribution function of the standardized normal distribution)

standard deviation:

\(u\left( \widehat{y} \right) = \sqrt{u^{2}(y) - \left( \widehat{y} - y \right) \cdot \widehat{y}}\)

This guarantees that \(\widehat{y}\) always will have a positive value.

The probabilistic symmetric coverage interval

For the result value \(y\) and the standard uncertainty \(u(y)\), this interval is given by the two limits:

\(y^{\vartriangleleft} = y - k_{p}u(y)\) with \(p = \omega(1 - \frac{\gamma}{2})\)

\(y^{\vartriangleright} = y + k_{p}u(y)\) with \(q = 1 - \frac{\omega\gamma}{2})\)

A normal distribution is assumed therein which is cut at the left side at the value of zero.

The shortest coverage interval

This interval, no longer be symmetric, is defined by the limits:

\(y^{<} = y + k_{p}u(y)\) with \(p = (1 + \omega(1 - \gamma))/2\)

\(y^{>} = y - k_{p}u(y)\)

These are modified In the case of \(y^{<} < 0\):

\(y^{<} = 0\)

\(y^{>} = y + k_{q}u(y)\) with \(q = 1 - \omega\gamma\)

Numerical estimation of these interval limits in the case of a MC-Simulation

From a MC-Simulation of a physical quantity, an array of about \(N =\)10⁴ through 10⁶ simulated values \(y_{i}\) results which is sorted ascending. A simple relation between a value \(y_{i}\) and the associated probability value \(p_{i}\) holds:

\(p_{i} = \frac{i}{N}\) with \(\sum_{i = 1}^{N}{p_{i} = 1}\)

For a given probability \(p_{x}\) its associated index \(k_{x}\) within the array is found by:

\(k_{x} = p_{x}N\) .

The associated \(y\) value of the interval limit, considered as a quantile lies between the two adjacent values \(y_{int(k_{x})}\) und \(y_{int\left( k_{x} \right) + 1}\) .

The shortest coverage interval (min_length) corresponding to the probability \((1 - \gamma)\) is searched for within a simple loop over the array y(i) as follows:

imax = gamma*N
min_length = 1E+20
do i=1,imax
    q_left = y(i)
    q_right = y(int(N * (1 - i/N)))
    if (q_right - q_left < min_length) then
        min_length = q_right - q_left
        q_left_min = q_left
        q_right_min = q_right
    endif
enddo