  We compare the accuracy, precision and reliability of different methods for
estimating key system parameters for two-level systems subject to Hamiltonian
evolution and decoherence. It is demonstrated that the use of Bayesian
modelling and maximum likelihood estimation is superior to common techniques
based on Fourier analysis. Even for simple two-parameter estimation problems,
the Bayesian approach yields higher accuracy and precision for the parameter
estimates obtained. It requires less data, is more flexible in dealing with
different model systems, can deal better with uncertainty in initial conditions
and measurements, and enables adaptive refinement of the estimates. The
comparison results shows that this holds for measurements of large ensembles of
spins and atoms limited by Gaussian noise as well as projection noise limited
data from repeated single-shot measurements of a single quantum device.
