How accurately can parameters from exponential models be estimated? A Bayesian view

Estimating the amplitudes and decay rate constants of exponentially decaying signals is an important problem in NMR. Understanding how the uncertainty in the parameter estimates depends on the data acquisition parameters and on the "true" but unknown values of the exponential signal parameters is an important step in designing experiments and determining the amount and quality of the data that must be gathered to make good parameter estimates. In this article, Bayesian probability theory is applied to this problem. Explicit relationships between the data acquisition parameters and the "true" but unknown exponential signal parameters are derived for the cases of data containing one and two exponential signal components. Because uniform prior probabilities are purposely employed, the results are broadly applicable to experimental parameter estimation.