Calculation of the best estimates for measurements of radioactive substances when the presence of the analyte is not assured

Abstract

The best estimate and its standard deviation are calculated for the case when the a priori probability that the analyte is absent from the test sample is not zero. In the calculation, a generalization of the Bayesian prior that is used in the ISO 11929 standard is applied. The posterior probability density distribution of the true values, given the observed value and its uncertainty, is a linear combination of the Dirac delta function and the normalized, truncated, normal probability density distribution defined by the observed value and its uncertainty. The coefficients of this linear combination depend on the observed value and its uncertainty, as well as on the a priori probability. It is shown that for a priori probabilities larger than zero the lower level of the uncertainty interval of the best estimate reaches the unfeasible range (i.e., negative activities). However, for a priori probabilities in excess of 0.26 it reaches the unfeasible range even for positive observed values. The upper limit of the confidence interval covering a predefined fraction of the posterior is derived.

Appendix: Derivation of the posterior

The prior, given in Eq. (1), yields with the likelihood function defined with the measurement result x ± σ _x , the posterior

$$p^{\prime} (a\left| x \right.) = C \, \left[ {p_{0}\, \delta (a/\sigma_{x} ) + (1 - p_{0} )\, \text{H}(a/\sigma_{x} )} \right] \, \varphi \left( {(a - x)/\sigma_{x} } \right),$$

(10)

where φ((a − x)/σ _x ) denotes the normal distribution described by the measurement result

$$\varphi \left( {(a - x)/\sigma_{x} } \right) = \frac{{\text{exp}\left( {{ - \frac{{(a - x)^{2} }}{{2\sigma_{x}^{2} }}}}\right) }}{{\sqrt {2\pi } \sigma_{x} }}.$$

(11)

It should be observed that the arguments in the prior were rescaled by σ ⁻¹ _x because the Dirac delta function and the Heaviside function are functions of a dimensionless parameter. Whereas the rescaling has no effect on the Heaviside function since H(a) = H(a/σ _x ), the Dirac delta function is normalized to unity, i.e., ∫δ(x)dx = 1, from where it follows ∫δ(a/σ _x )da = σ _x .

The constant C follows from the normalization condition for the posterior

$$\int\limits_{0}^{\infty } {p^{\prime} (a\left| x \right.) \, {\text{d}}a = C\int\limits_{0}^{\infty } {\left[ {p_{0} \, \delta (a/\sigma_{x} ) + (1 - p_{0} ) \, \text{H}(a/\sigma_{x} )} \right]} \, \varphi \left( {(a - x)/\sigma_{x} } \right) \, {\text{d}}a = 1.}$$

(12)

Since f(x,a) δ(a) = f(x,0) δ(a) for the normalization constant follows

$$C = \frac{1}{{p_{0}\, \sigma_{x} \, \varphi (x/\sigma_{x} ) + (1 - p_{0} ) \, \Phi (x/\sigma_{x} )}},$$

(13)

where Φ(x/σ _x ) denotes the cumulative function of the standardized normal distribution. Then it is easy to see that the posterior is also a linear combination of the Dirac delta function and a normal distribution truncated at zero. The coefficients of this linear combination are

$$\alpha = \frac{{p_{0} \;\varphi (x/\sigma_{x} )}}{{p_{0} \;\sigma_{x} \;\varphi (x/\sigma_{x} ) + (1 - p_{0} )\;\Phi (x/\sigma_{x} )}}$$

(14)

and

$$\beta = \frac{{(1 - p_{0} ) \, \Phi (x/\sigma_{x} )}}{{p_{0} \, \sigma_{x} \, \varphi (x/\sigma_{x} ) + (1 - p_{0} ) \, \Phi (x/\sigma_{x} )}}$$

(15)

for the Dirac delta function and the truncated normal distribution, respectively.