  The $\mu$Dose system was developed to allow the measurement of environmental
levels of natural radioactive isotopes. The system records $\alpha$ and $\beta$
particles along with four decay pairs arising from subsequent decays of
$^{214}$Bi/$^{214}$Po, $^{220}$Rn/$^{216}$Po, $^{212}$Bi/$^{212}$Po and
$^{219}$Rn/$^{215}$Po. Under the assumption of secular equilibrium this allows
to assess the specific radioactivities of $^{238}$U, $^{235}$U, $^{232}$Th
decay chains and $^{40}$K. This assessment provides results with uncertainties
which are correlated and, thus, require the development of an error estimation
methodology which considers this issue. Here we present two different
approaches for uncertainty propagation based on Monte Carlo and Bayesian
methods. Both approaches produce statistically indistinguishable results and
allow significantly better dose rate precision than when the correlations are
not accounted for. In the given example, the dose rate precision is improved by
a factor of two.
